Quartic function
The quartic function is the bridge between the cubic function and more advanced functions such as the quintic and sextic.
Objective
- Present quartic function and quartic equation.
- Introduce the concept of roots of equal absolute value.
- Show how to predict and calculate equal roots, techniques that will be useful when applied to higher order functions.
- Simplify the depressed quartic.
- Show that the quartic equation is effectively solved when at least one root is known.
- Present the "resolvent" cubic function.
- Show how to derive and use the quartic formula.
Lesson
Introduction
The quartic function is the sum of powers of from through :
usually written as:
If the function becomes
Within this page we'll say that:
- both coefficients must be non-zero,
- coefficient must be positive (simply for our convenience),
- all coefficients must be real numbers, accepting that the function may contain complex roots.
The quartic equation is the quartic function equated to zero:
.
Roots of the function are values of that satisfy the quartic equation. Template:RoundBoxTop Because the function is "quartic" (maximum power of is ), the function contains exactly roots, an even number of complex roots and an even number of real roots.
Other combinations of real and complex roots are possible, but they produce complex coefficients. Template:RoundBoxBottom Template:RoundBoxTop
The figure shows a typical quartic function.
The function crosses the
axis in 4 different places. The function has 4 roots:
This function contains one local minimum, one local maximum and one absolute minimum. There is no absolute maximum.
Because the function contains one absolute minimum:
- If abs() is very large, is always positive.
- If absolute minimum is above axis, curve does not cross axis and function contains only complex roots.
- There is always at least one point where the curve is parallel to axis.
The curve is never parallel to the axis. For any real value of there is always a real value of
Template:RoundBoxBottom
Template:RoundBoxTop
If coefficient is missing, the quartic function becomes and
For a stationary point
When coefficient is missing, there is always a stationary point at Template:RoundBoxBottom
Because coefficient is missing, there is a stationary point where
If coefficients are missing, the quartic function becomes a quadratic in
The curve (red line) in diagram has equation:
The quartic equation may be solved as: where or
or
The quartic function may be expressed as
Unless otherwise noted, references to "quartic function" on this page refer to function of form Template:RoundBoxBottom
There is no absolute minimum.
Coefficient may be negative as shown in diagram.
As abs increases, the value of is dominated by the term
When abs is very large, is always negative.
Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient
positive.
Template:RoundBoxBottom
Sum of roots is
axis compressed for clarity.
When sum of roots is coefficient
In the diagram, roots of are
Sum of roots
Therefore coefficient Template:RoundBoxBottom
Function as product of linear function and cubic
When is a root of the function, the function may be expressed as:
where
When one real root is known, the other three roots may be calculated as roots of the cubic function
In the diagram the quartic function has equation:
It is known that is a root of this function.
The associated cubic has equation:
The 2 curves coincide at points the three points that are roots of both functions. Template:RoundBoxBottom
Function defined by 5 points
Any 5 points on the curve may be used to define the function.
Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function.
See Figure 1. The quartic function may be defined by any 5 unique points on the curve.
For example, let us choose the five points:
Rearrange the standard quartic function to prepare for the calculation of
For function solveMbyN see
"Solving simultaneous equations" .
# python code
points = (-5,0), (-2,0), (1,0), (3,-6), (6,2)
L11 = []
for point in points :
x,y = point
L11 += [[x*x*x*x, x*x*x, x*x, x, 1, -y]]
print (L11)
[[ 625.0, -125.0, 25.0, -5.0, 1.0, 0.0], #
[ 16.0, -8.0, 4.0, -2.0, 1.0, 0.0], #
[ 1.0, 1.0, 1.0, 1.0, 1.0, 0.0], # matrix supplied to function solveMbyN() below.
[ 81.0, 27.0, 9.0, 3.0, 1.0, 6.0], # 5 rows by 6 columns.
[1296.0, 216.0, 36.0, 6.0, 1.0, -2.0]] ## python code
output = solveMbyN(L11)
print (output)
# 5 coefficients a, b, c, d, e:
(0.02651515151515152, 0.004545454545454542, -0.847727272727273, -0.728787878787879, 1.5454545454545459)Template:RoundBoxBottom Template:RoundBoxBottom Quartic function defined by the 5 points is Template:RoundBoxBottom
Function defined by 3 points and 2 slopes
Any 3 points on the curve and the slopes at any 2 of these points may be used to define the function.
Slope at slope at
Because the quartic function contains 5 coefficients, 5 simultaneous equations are needed to define the function.
See Figure 1. The quartic function may be defined by any 3 unique points on the curve and the slopes at any 2 of these points.
For example, let us choose the three points:
At point slope is
At point slope is
Rearrange the standard quartic function to prepare for the calculation of
Rearrange the standard cubic function of slope to prepare for the calculation of
For function solveMbyN see
"Solving simultaneous equations" .
# python code
def makeEntry(input) :
x,y,s = ( tuple(input) + (None,) )[:3]
L1 = []
if s != None :
L2 = [ float(v) for v in [4*x*x*x, 3*x*x, 2*x, 1, 0, -s] ]
L1 += [ L2 ]
L2 = [ float(v) for v in [x*x*x*x, x*x*x, x*x, x, 1, -y]]
L1 += [ L2 ]
return L1
t1 = (
(-2,-2, 0), # point (-2, -2) with slope 0.
(6,-4, 0), # point (6, -4) with slope 0.
(4,1), # point (4, 1)
)
L1 = []
for v in t1 : L1 += makeEntry ( v )
print (L1)
[[ -32.0, 12.0, -4.0, 1.0, 0.0, 0.0], #
[ 16.0, -8.0, 4.0, -2.0, 1.0, 2.0], #
[ 864.0, 108.0, 12.0, 1.0, 0.0, 0.0], # matrix supplied to function solveMbyN() below.
[1296.0, 216.0, 36.0, 6.0, 1.0, 4.0], # 5 rows by 6 columns.
[ 256.0, 64.0, 16.0, 4.0, 1.0, -1.0]]. ## python code
output = solveMbyN(L1)
print (output)
# 5 coefficients a, b, c, d, e:
(0.03255208333333339, -0.2526041666666665, -0.3072916666666667, 2.84375, 2.375)Template:RoundBoxBottom Template:RoundBoxBottom Quartic function defined by three points and two slopes is:
Associated cubic functions
When p == -2
p = -2.
In this case roots of associated cubic include x = p.
Quartic function is:
When associated cubic function is :
Three blue vertical lines show 3 values of where and
In this case roots of include Template:RoundBoxBottom
When p == 5
p = 5.
In this case the one root of associated cubic excludes x = p.
Quartic function is:
When associated cubic function is :
Two blue vertical lines show 2 values of where
In this case the one root of excludes Template:RoundBoxBottom
When p == 6
p = 6.
In this case the one root of associated cubic includes x = p.
Quartic function is:
When associated cubic function is:
One blue vertical line shows 1 value of where
In this case the one root of includes Template:RoundBoxBottom Template:RoundBoxBottom Template:RoundBoxBottom
Examples
Quartic with 2 stationary points
In the diagram the red line represents quartic function
The grey line is the first derivative of
The 2 roots of and show that has stationary points at and Template:RoundBoxBottom
Quartic with 1 stationary point
In the diagram the red line represents quartic function
The grey line is the first derivative of
The 1 root of (approx.), shows that has 1 stationary point where Template:RoundBoxBottom Template:RoundBoxBottom Template:RoundBoxBottom
First and second derivatives
Points of inflection
Dotted portion of black line shows where f(x) is always concave down.
Dotted portion of red line shows where g(x) is decreasing.
Dotted portion of blue line shows where h(x) is negative.
When h(x) is negative, f(x) is concave down.
When h(x) is positive, f(x) is concave up.
In the diagram the black line has equation:
The first derivative, the red line, has equation:
The second derivative, the blue line, has equation:
When
- is increasing.
- is positive.
- is always concave up.
When
- is at a local maximum.
- Concavity of is between up and down.
When
- is decreasing.
- is negative.
- is always concave down.
When
- is at a local minimum.
- Concavity of is between down and up.
When
- is increasing.
- is positive.
- is always concave up.
Template:RoundBoxTop
The roots of
Let point on have coordinates
Let point on have coordinates
At point concavity of changes from up to down.
At point concavity of changes from down to up.
The points (the coordinates of which are roots of ) are the points of inflection of Template:RoundBoxBottom Template:RoundBoxBottom
Maxima and minima
Point on is a stationary point. at point is concave down. Point is local maximum.
Point on is a stationary point. at point is concave up. Point is local minimum.
Similarly, point is local minimum.
In the diagram the black line has equation:
The first derivative, the red line, has equation:
The second derivative, the blue line, has equation:
Roots of
Let point on have coordinates
At is positive. Point is a stationary point and at is concave up. Point is a local minimum.
Let point on have coordinates
At is negative. Point is a stationary point and at is concave down. Point is a local maximum.
Let point on have coordinates
At is positive. Point is a stationary point and at is concave up. Point is a local minimum. Template:RoundBoxBottom
Quartic with 2 stationary points
Black line:
Red line:
Blue line:
Dotted portion of black line shows where is concave down.
In the diagram, point on has coordinates
Similarly, points have coordinates
has roots:
Points are stationary points.
has roots:
Points are points of inflection.
At point is positive. at is concave up.
Point is local minimum.
Template:RoundBoxTop
Summary:
- Point is both stationary point and point of inflection.
- Point is point of inflection.
- Point is both stationary point and local minimum.
Template:RoundBoxBottom Template:RoundBoxBottom Template:RoundBoxBottom
The simplest quartic function
Point is a root of
Point is a root of
Point is intercept of
The simplest quartic function has coefficients
Red line in diagram has equation:
First derivative (not shown) of
When There is a stationary point on when point
Second derivative (not shown) of
When There is a point of inflection on when
For every non-zero value of is positive. To left and right of point is always concave up. Point is both local minimum and absolute minimum.
- Point is stationary point and point of inflection and absolute minimum.
Curve is useful for finding the fourth root of a real number.
Solve:
This is equivalent to finding a root of function
If you use Newton's method to find a root of this would be more efficient than solving Template:RoundBoxBottom
Roots of equal absolute value
Template:RoundBoxTop The standard quartic function:
For in substitute Call this
For in substitute Call this
Combine and to eliminate and produce an equation in
We are interested in coefficient of
If is a solution and function has 2 roots of form where
An example:
Template:RoundBoxTop
In the diagram the red line has equation:
has roots of equal absolute value.
The 2 roots of equal absolute value are: Template:RoundBoxBottom The method works with complex roots of equal absolute value: Template:RoundBoxTop
In the diagram the red line has equation:
has roots of equal absolute value.
The 2 roots of equal absolute value are: Template:RoundBoxBottom Template:RoundBoxBottom
Equal roots
Template:RoundBoxTop Equal roots occur when the function and the slope of the function both equal zero.
Begin the process of reducing to linear functions.
Combine to produce 2 cubic functions:
where:
where:
Combine to produce 2 quadratic functions:
where:
where:
Combine to produce 2 linear functions:
where:
where:
From
From
If
The value is in fact:
+ 2048aaaaacddeeee - 768aaaaaddddeee - 1536aaaabcdddeee + 576aaaabdddddee
- 1024aaaacccddeee + 1536aaaaccddddee - 648aaaacdddddde + 81aaaadddddddd
+ 1152aaabbccddeee - 480aaabbcddddee + 18aaabbdddddde - 640aaabcccdddee
+ 384aaabccddddde - 54aaabcddddddd + 128aaacccccddee - 80aaaccccdddde
+ 12aaacccdddddd - 216aabbbbcddeee + 81aabbbbddddee + 144aabbbccdddee
- 86aabbbcddddde + 12aabbbddddddd - 32aabbccccddee + 20aabbcccdddde
- 3aabbccddddddwhich, by removing values (common to all values), may be reduced to:
status = (
+ 2048aaaceeee - 768aaaddeee - 1536aabcdeee + 576aabdddee
- 1024aaccceee + 1536aaccddee - 648aacdddde + 81aadddddd
+ 1152abbcceee - 480abbcddee + 18abbdddde - 640abcccdee
+ 384abccddde - 54abcddddd + 128acccccee - 80accccdde
+ 12acccdddd - 216bbbbceee + 81bbbbddee + 144bbbccdee
- 86bbbcddde + 12bbbddddd - 32bbccccee + 20bbcccdde
- 3bbccdddd
)If there are at least 2 equal roots which may be calculated as shown below. Template:RoundBoxTop If coefficient is non-zero, it is not necessary to calculate
If coefficient verify that before proceeding. Template:RoundBoxBottom
No equal roots
Red line in diagram is of function:
There are no equal roots.
Template:RoundBoxBottom
Exactly 2 equal roots
Red line in diagram is of function:
There are 2 equal roots at
Template:RoundBoxTop The following 3 graphs show the steps that lead to calculation of equal roots at point
-
Figure 1. graph of and 2 associated cubic functions.
-
Figure 2. graph of and 2 associated quadratic functions.
-
Figure 3. graph of and 2 associated linear functions.
In all graphs, all curves have a common root at point Template:RoundBoxBottom
Template:RoundBoxTop See "Function as product of linear function and cubic" above.
To calculate all roots:
# python code.
a,b,c,d,e = 1,6,-48,-182,735
# The associated cubic:
p = -7
A = a
B = A*p + b
C = B*p + c
D = C*P + d
# The associated quadratic:
a1 = A
b1 = a1*p + B
c1 = b1*p + C
a1,b1,c1
(1, -8, 15)Roots of quadratic function are
All roots of are Template:RoundBoxBottom Template:RoundBoxBottom
Exactly 3 equal roots
Red line in diagram is of function:
In this case the calculation of is not appropriate because there are more than 2 equal roots. Try equations Both of these are equivalent to: blue line in diagram.
Discriminant of has two equal roots at Therefore has 3 equal roots at Template:RoundBoxBottom
Four equal roots
Red line in diagram is of function:
In this case are all null.
This is the only case in which are null.
are both equivalent to: blue line in diagram.
has one root at Therefore has 4 equal roots at Template:RoundBoxBottom
Two pairs of equal roots
Red line in diagram is of function:
In this case are both null.
are both equivalent to: blue line in diagram.
has one root at and one root at Therefore has 2 equal roots at and 2 equal roots at Template:RoundBoxTop
This method is valid for complex roots.
For example:
In this case are both null.
are both equivalent to: blue line in diagram.
Roots of are
has 2 roots equal to and 2 roots equal to Template:RoundBoxBottom Template:RoundBoxBottom
Summmary
| No equal roots | 2 equal roots | 3 equal roots | 4 equal roots | 2 pairs of equal roots | |
|---|---|---|---|---|---|
| Cubic: 1(a), 2(a) | different | different | different | same | different |
| Quadratic: 1(b), 2(b) | different | different | same, 1root | null | same, 2roots |
| Linear: 1(c), 2(c) | different | same | null | null | null |
Caution
In this example, method calculates 2 legitimate equal roots at and 2 false equal roots at
Black line in diagram has equation:
is a quartic function with exactly 2 equal roots and coefficient missing.
Calculation of equal roots of gives linear functions null and quadratic functions
with equal roots of
Usually, this indicates that should have 2 equal roots at and 2 equal roots at
It is obvious that is not a root of
When slope of derivative
Value of
This example indicates that it would be wise to verify that calculated equal roots are in fact valid roots of
Template:RoundBoxBottom
Template:RoundBoxBottom
Depressed quartic
Template:RoundBoxTop A depressed quartic is any quartic function with any one or more of coefficients missing. Within this section a depressed quartic has coefficient missing.
To produce the depressed quartic:
Let Substitute in expand and simplify:
where:
When equated to becomes the depressed equation:
Be prepared for the possibility that any 1 or more of may be zero.
Coefficient B missing
If coefficient becomes a quadratic in
has the appearance of a quadratic.
The black line:
The red line:
where
The grey line:
- Absolute minima of and of and point of inflection of occur where
- is always positive. is always concave up.
Template:RoundBoxTop If contains 2 pairs of equal roots, coefficient
The converse is not necessarily true.
If contains 4 equal roots, coefficients
Template:RoundBoxBottom
Template:RoundBoxBottom
Coefficient C missing
axis compressed for clarity.
If coefficient becomes:
in which case is a solution and is a root of
Curve (red line) in example has equation:
Coefficients of depressed function are:
# python code
a,b,c,d,e = 8, 16, 24, 89, 40
A = 16*a*c - 6*b*b
B = 64*a*a*d - 32*a*b*c + 8*b*b*b
C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b
A,B,C
(1536, 299008, 0)Coefficient of depressed function is missing. is a solution.
Using
one root of Template:RoundBoxBottom
Resolvent cubic
Template:RoundBoxTop This section introduces a special cubic function called "resolvent" because it helps to resolve a requirement, the calculation of the roots of the quartic.
The depressed quartic:
For substitute
For substitute
Simplify
Simplify
From
In replace with expand, simplify, gather like terms and result is:
or Template:RoundBoxTop where:
Template:RoundBoxBottom Template:RoundBoxTop From Template:RoundBoxBottom Template:RoundBoxTop Some simple changes reduce the number of calculations and also the sizes of coefficients
where
where
Then:
Divide all 4 coefficients by
A close examination of coefficients shows that both coefficients are always
exactly divisible by
Therefore, all four coefficients may be defined as follows:
Template:RoundBoxBottom Template:RoundBoxBottom Template:RoundBoxBottom
Solving quartic equation
This section presents 4 examples that show how to use the depressed quartic and the resolvent cubic to solve the quartic equation.
Four real roots
Resolvent cubic contains 3 real, positive roots.
Calculate roots of: Template:RoundBoxBottom
Calculate coefficients of depressed quartic:
a,b,c,d,e = 1,-1,-19,-11,30
A = 16*a*c - 6*b*b
B = 64*a*a*d - 32*a*b*c + 8*b*b*b
C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b
print (A,B,C)
-310 -1320 6669Template:RoundBoxBottom Calculate coefficients of resolvent cubic:
P = 64
Q = 32*A
R = 4*A*A - 16*C
S = -B*B
print (P,Q,R,S)
64 -9920 277696 -1742400Template:RoundBoxBottom Calculate roots of cubic function:
There are 3 real, positive roots:
Using 3 roots of calculate 4 roots of
# python code
for U in (9, 25, 121) :
print ('\nU =', U)
sqrtU = U ** 0.5
for u in (sqrtU, -sqrtU) :
V = -(A/2 + U) - B/(4*u)
v = V ** .5
for t in (u+v, u-v) :
x = (-b+t) / (4*a)
y = a*x**4 + b*x**3 + c*x**2 + d*x + e
print ('x:',x, '; y:',y)
U = 9
x: 5.0 ; y: 0.0
x: -3.0 ; y: 0.0
x: 1.0 ; y: 0.0
x: -2.0 ; y: 0.0
U = 25
x: 5.0 ; y: 0.0
x: -2.0 ; y: 0.0
x: 1.0 ; y: 0.0
x: -3.0 ; y: 0.0
U = 121
x: 5.0 ; y: 0.0
x: 1.0 ; y: 0.0
x: -2.0 ; y: 0.0
x: -3.0 ; y: 0.0Roots of are: Template:RoundBoxBottom All 3 values of produce the same results, but not in same sequence.
It is not necessary to calculate all 3 roots of resolvent cubic. Any one non-zero root is sufficient to do the job.
Two real and two complex roots
Example 1
Calculate roots of: Template:RoundBoxBottom
Template:RoundBoxTop Calculate coefficients of depressed quartic:
a,b,c,d,e = 1,2,18,-70,-87
A = 16*a*c - 6*b*b
B = 64*a*a*d - 32*a*b*c + 8*b*b*b
C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b
print (A,B,C)
264 -5568 -12208Template:RoundBoxBottom Calculate coefficients of resolvent cubic:
P = 64
Q = 32*A
R = 4*A*A - 16*C
S = -B*B
print (P,Q,R,S)
64 8448 474112 -31002624Template:RoundBoxBottom Calculate one real root of cubic function:
is one real root. Choose
Calculate roots of
# python code
U = 36
u1 = U**.5
for u in (u1, -u1) :
V = -(A/2 + U) - B/(4*u)
if V >= 0 : v = V**.5
else : v = 1j * (-V)**.5
for t in (u+v, u-v) :
x = (-b+t)/(4*a)
# Check result. Expecting sum = 0.
sum = a*x**4 + b*x**3 + c*x**2 + d*x + e
print ('x =',x, '; sum =',sum )
x = 3.0 ; sum = 0.0
x = -1.0 ; sum = 0.0
x = (-2+5j) ; sum = 0j
x = (-2-5j) ; sum = 0jTemplate:RoundBoxBottom Template:RoundBoxBottom
Example 2
Calculate roots of: Template:RoundBoxBottom Calculate coefficients of depressed quartic:
a,b,c,d,e = 3, -6, -41, 44, -189
A = 16*a*c - 6*b*b
B = 64*a*a*d - 32*a*b*c + 8*b*b*b
C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b
print (A,B,C)
-2184, 0, -1229040Notice that coefficient Template:RoundBoxBottom Calculate coefficients of resolvent cubic:
P = 64
Q = 32*A
R = 4*A*A - 16*C
S = -B*B
print (P,Q,R,S)
1, -1092, 605376, 0Notice that coefficient Template:RoundBoxBottom Calculate roots of cubic function:
Roots are
Value cannot be used because it will cause error Divide by zero
at statement V = -(A/2 + U) - B/(4*u).
Calculate roots of
# python code
U = 546+554.3103823671355j
print ('\nU =',U)
sqrtU = U ** 0.5
for u in (sqrtU, -sqrtU) :
V = -(A/2 + U) - B/(4*u)
v = V ** 0.5
s1 = '\nu,v'
print (s1,eval(s1))
for t in (u+v, u-v) :
x = (-b+t)/(4*a)
# Check result. Expecting sum = 0.
sum = a*x**4 + b*x**3 + c*x**2 + d*x + e
print ('x =', x,'; sum =',sum)
U = (546+554.3103823671355j)
u,v ((25.729935131257832+10.771701901683684j), (25.729935131257832-10.771701901683684j))
x = (4.788322521876306 + 0j) ; sum = (1.9895196601282805e-13 + 0j)
x = (0.5 + 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j)
u,v ((-25.729935131257832-10.771701901683684j), (25.729935131257832-10.771701901683684j))
x = (0.5 - 1.795283650280614j) ; sum = (5.684341886080802e-14 + 0j)
x = (-3.7883225218763052 + 0j) ; sum = (1.7053025658242404e-13 + 0j)Values of are: Template:RoundBoxBottom
Depressed quartic as quadratic
In this example coefficient of depressed quartic
Therefore, resolvent cubic can be ignored and depressed quartic processed as quadratic in
where
Solutions of this quadratic are:
T1,T2 = 2648.1182474349434, -464.11824743494344
t1 = T1 ** 0.5; t2 = ((-T2) ** 0.5) * 1j
for t in (t1,-t1,t2,-t2) :
x = (-b+t)/(4*a)
# Check result. Expecting sum = 0.
sum = a*x**4 + b*x**3 + c*x**2 + d*x + e
print ('x =', x,'; sum =',sum)
x = 4.788322521876305 ; sum = -1.7053025658242404e-13
x = -3.788322521876305 ; sum = -1.7053025658242404e-13
x = (0.5 + 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j)
x = (0.5 - 1.7952836502806138j) ; sum = (-2.842170943040401e-14 + 0j)or
x = 0.5 ± 4.288322521876305, 0.5 ± 1.7952836502806138jWith precision of 15, values of are same as those shown above.
When roots of quartic function are of form p ± q, p ± r,
coefficient of depressed function
Four complex roots
Calculate roots of: Template:RoundBoxBottom
Calculate coefficients of depressed quartic:
4128 344064 9683200Template:RoundBoxBottom Calculate coefficients of resolvent cubic:
64 132096 -86769664 -118380036096Template:RoundBoxBottom Calculate one root of cubic function:
There are 3 real roots: Choose
Negative is chosen here to show that any 1 of the roots produces the correct result.
Calculate roots of
# python code
U = -784
u1 = 1j * (-U)**.5
for u in (u1, -u1) :
V = -(A/2 + U) - B/(4*u)
v = V**.5
for t in (u+v, u-v) :
x = (-b+t)/(4*a)
# Check result. Expecting sum = 0.
sum = a*x**4 + b*x**3 + c*x**2 + d*x + e
print ('x =', x,'; sum =',sum)
# python expresses complex numbers with 'j'.
x = (13+19j) ; sum = 0j
x = (-3-5j) ; sum = 0j
x = (13-19j) ; sum = 0j
x = (-3+5j) ; sum = 0jQuartic formula
The substitutions made above can be used to produce a formula for the solution of the quartic equation.
See main articles "The general case" or "General formula for roots." Template:RoundBoxTop Both links above point to formula for equation Template:RoundBoxBottom Given quartic equation: calculate the 4 values of
where:
Template:RoundBoxTop Coefficients of depressed quartic:
Template:RoundBoxTop Coefficients of resolvent cubic:
Template:RoundBoxTop Coefficients of depressed cubic:
Template:RoundBoxBottom Template:RoundBoxTop One root of resolvent cubic:
may be negative.
Template:RoundBoxTop One root of quartic:
may be positive or negative.
may be positive or negative.
Template:RoundBoxBottom Formula above produces one value of Python code below utilizes and to produce 4 values of and then, four values of
An example:
axis compressed for clarity.
Calculate roots of
# Python code.
a,b,c,d,e = 4, 4, -75, -776, -1869
values_of_t = [
t
# Coefficients of depressed quartic:
for A in (16*a*c - 6*b*b,)
for B in (64*a*a*d -32*a*b*c + 8*b*b*b,)
for C in (256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b,)
# Coefficients of resolvent cubic:
for a1 in (64,)
for b1 in (32*A,)
for c1 in (4*A*A - 16*C,)
for d1 in (-B*B,)
for U in [
# The resolvent cubic:
(-b1+t1)/(3*a1)
# Coefficients of depressed resolvent cubic:
for A1 in (9*a1*c1 - 3*b1*b1,)
for B1 in (27*a1*a1*d1 - 9*a1*b1*c1 + 2*b1*b1*b1,)
# One root of resolvent cubic:
for C1 in (-A1/3,)
for Δ in (B1*B1 - 4*C1*C1*C1,)
for δ in (Δ**0.5,)
for W in ((-B1 + δ)/2,)
for w in (W**(1/3),)
for t1 in (w + C1/w,) # See note below.
]
# Prepare to calculate 4 values of t.
for u1 in (U**.5,)
for v1 in ( -(A/2 + U) ,)
# Calculate 4 values of t.
for u in (u1, -u1,)
for V in ( v1 - B/(4*u),)
for v in (V**.5,)
for t in (u+v, u-v)
]
print ('values_of_t =', values_of_t)
values_of_t = [116, -44, (-36+64j), (-36-64j)]Because is a depressed quartic function, sum of four values_of_t
Template:RoundBoxBottom
# Python code.
# Calculate 4 separate roots.
values_of_x = [
(-b + t)/(4*a)
for t in values_of_t
]
print ('values_of_x =', values_of_x)
values_of_x = [7, -3, (-2.5 + 4j), (-2.5 - 4j)]
In python the imaginary part of a complex number is shown with instead of
If contains 4 equal roots and
If contains 3 or more equal roots, statement for t1 in (w + C1/w,) fails with divisor
Before using this formula, check for equal roots as in "Exactly 3 equal roots" above. Template:RoundBoxTop Values displayed above have been edited slightly. Actual calculated values were:
values_of_x = [7.000000000000001,
-3.0000000000000044,
(-2.499999999999998+4.000000000000001j),
(-2.4999999999999987-4.000000000000001j)]Template:RoundBoxBottom Template:RoundBoxBottom
In practice
Template:RoundBoxTop
The following Python code implements the quartic formula. However, under statement if B4 == 0 :
there is code that processes the depressed quartic as a quadratic in
This ensures that execution of formula does not fail with error Divide by zero
at statement for V in ( v1 - B4/u,).
Template:RoundBoxTop
# python code
import cmath
cxSqrt = cmath.sqrt # Square root of complex number.
def rootsOfQuartic (abcde) :
'''
x1,x2,x3,x4 = rootsOfQuartic ((a,b,c,d,e))
Each member of input must be int or float or Decimal object.
Int or Decimal object in input is quietly converted to float.
Output may be None.
'''
def formatResults (x1x2x3x4) :
'''
This function improves appearance of results.
(8 + 0j) becomes 8.0
'''
values_of_x = list (x1x2x3x4)
for p in (0,1,2,3) :
v = values_of_x[p]
if isinstance (v, complex) and (v.imag == 0) : values_of_x[p] = v.real
return values_of_x
status = 0
try : a,b,c,d,e = [ float(v) for v in abcde ]
except : status = 1
if status :
print ('rootsOfQuartic () 1: Error creating coefficients a,b,c,d,e.')
return None
if a == 0 :
print ('rootsOfQuartic () 2: Coefficient a must be non-zero.')
return None
# Coefficients of depressed quartic, modified.
# A = 16*a*c - 6*b*b
A2 = 8*a*c - 3*b*b
# B = 64*a*a*d - 32*a*b*c + 8*b*b*b
B4 = 16*a*a*d - 8*a*b*c + 2*b*b*b
C = 256*a*a*a*e - 64*a*a*b*d + 16*a*b*b*c - 3*b*b*b*b
if B4 == 0 :
# B = 0.
# Result returned from this section is type tuple, indicating that coefficient B4 = 0.
if A2==C==0 :
# 4 equal roots.
root = -b/(4*a)
return tuple(formatResults((root,root,root,root)))
# t**4 + At**2 + (0)t + C
# Depressed quartic is quadratic in T:
# T**2 + AT + C where T = t**2
# T**2 + 2(A2)T + C where A = 2(A2)
# -2(A2) +/- (4(A2)(A2) - 4C)**0.5
# T = -------------------------------- = -A2 +/- ((A2)(A2) - C)**0.5
# 2
disc = A2*A2 - C
if disc >= 0 : root = disc ** 0.5
else : root = ((-disc) ** 0.5) * 1j
T1 = -A2 - root ; T2 = -A2 + root
t1 = cxSqrt(T1) ; t2 = cxSqrt(T2)
values_of_t = (t1,-t1,t2,-t2)
values_of_x = [ (-b + t)/(4*a) for t in values_of_t ]
return tuple(formatResults(values_of_x))
# B4 is non-zero. Therefore, all of (S, U, u) are non-zero.
P,Q,R,S = 1, A2, (A2*A2 - C)/4, -B4*B4/4
# str1 = 'P,Q,R,S' ; print (str1, eval(str1))
U = oneRootOfCubic((P,Q,R,S)) # Resolvent cubic.
if U > 0 : sqrtU = U ** 0.5
elif U == 0 :
# This should not happen.
print ('rootsOfQuartic () 3: Internal error.')
return None
else : sqrtU = ((-U) ** 0.5) * 1j
v1 = -(A2+U)
values_of_t = [
t
for u in (sqrtU, -sqrtU)
for V in ( v1 - B4/u, )
for v in ( cxSqrt(V), )
for t in (u+v, u-v)
]
values_of_x = [
(-b + t)/(4*a)
for t in values_of_t
]
# Result returned from this section is type list, indicating that coefficient B4 != 0.
return formatResults(values_of_x)
For function oneRootOfCubic() see Cubic_function: In_practice.
Template:RoundBoxBottom
Examples
Template:RoundBoxTop
Python function equalRoots() below implements status as presented under
Equal roots above.
# python code
def equalRoots(abcde) :
'''
This function returns True if quartic function contains at least 2 equal roots.
'''
a,b,c,d,e = abcde
aa = a*a ; aaa = aa*a
bb = b*b ; bbb = bb*b ; bbbb = bb*bb
cc = c*c ; ccc = cc*c ; cccc = cc*cc ; ccccc = cc*ccc
dd = d*d ; ddd = dd*d ; dddd = dd*dd ; ddddd = dd*ddd ; dddddd = ddd*ddd
ee = e*e ; eee = ee*e ; eeee = ee*ee
v1 = (
+2048*aaa*c*eeee +576*aa*b*ddd*ee +1536*aa*cc*dd*ee +81*aa*dddddd
+1152*a*bb*cc*eee +18*a*bb*dddd*e +384*a*b*cc*ddd*e +128*a*ccccc*ee
+12*a*ccc*dddd +81*bbbb*dd*ee +144*bbb*cc*d*ee +12*bbb*ddddd
+20*bb*ccc*dd*e
)
v2 = (
-768*aaa*dd*eee -1536*aa*b*c*d*eee -1024*aa*ccc*eee -648*aa*c*dddd*e
-480*a*bb*c*dd*ee -640*a*b*ccc*d*ee -54*a*b*c*ddddd -80*a*cccc*dd*e
-216*bbbb*c*eee -86*bbb*c*ddd*e -32*bb*cccc*ee -3*bb*cc*dddd
)
return (v1+v2) == 0
t1 = (
((1, -1, -19, -11, 30), '4 unique, real roots.'),
((4, 4,-119, -60, 675), '4 unique, real roots, B4 = 0.'),
((1, 6, -48,-182, 735), '2 equal roots.'),
((1,-12, 50, -84, 45), '2 equal roots. B4 = 0.'),
((1,-20, 146,-476, 637), '2 equal roots, 2 complex roots.'),
((1,-12, 58,-132, 117), '2 equal roots, 2 complex roots. B4 = 0.'),
((1, -2, -36, 162, -189), '3 equal roots.'),
((1,-20, 150,-500, 625), '4 equal roots. B4 = 0.'),
((1, -6, -11, 60, 100), '2 pairs of equal roots, B4 = 0.'),
((4, 4, -75,-776,-1869), '2 complex roots.'),
((1,-12, 33, 18, -208), '2 complex roots, B4 = 0.'),
((1,-20, 408,2296,18020), '4 complex roots.'),
((1,-12, 83, -282, 442), '4 complex roots, B4 = 0.'),
((1,-12, 62,-156, 169), '2 pairs of equal complex roots, B4 = 0.'),
)
for (abcde, comment) in t1 :
print ()
fourRoots = rootsOfQuartic (abcde)
print (comment)
print (' Coefficients =', abcde)
print (' Four roots =', fourRoots)
print (' Equal roots detected:', equalRoots(abcde))
# Check results.
a,b,c,d,e = abcde
for x in fourRoots :
# To be exact, a*x**4 + b*x**3 + c*x**2 + d*x + e = 0
sum = (a*x**4 + b*x**3 + c*x**2 + d*x + e)
if sum :
# Create exception.
1/0
4 unique, real roots.
Coefficients = (1, -1, -19, -11, 30)
Four roots = [5.0, 1.0, -2.0, -3.0]
Equal roots detected: False
4 unique, real roots, B4 = 0.
Coefficients = (4, 4, -119, -60, 675)
Four roots = (2.5, -3.0, 4.5, -5.0)
Equal roots detected: False
2 equal roots.
Coefficients = (1, 6, -48, -182, 735)
Four roots = [5.0, 3.0, -7.0, -7.0]
Equal roots detected: True
2 equal roots. B4 = 0.
Coefficients = (1, -12, 50, -84, 45)
Four roots = (3.0, 3.0, 5.0, 1.0)
Equal roots detected: True
2 equal roots, 2 complex roots.
Coefficients = (1, -20, 146, -476, 637)
Four roots = [7.0, 7.0, (3+2j), (3-2j)]
Equal roots detected: True
2 equal roots, 2 complex roots. B4 = 0.
Coefficients = (1, -12, 58, -132, 117)
Four roots = ((3+2j), (3-2j), 3.0, 3.0)
Equal roots detected: True
3 equal roots.
Coefficients = (1, -2, -36, 162, -189)
Four roots = [3.0, 3.0, 3.0, -7.0]
Equal roots detected: True
4 equal roots. B4 = 0.
Coefficients = (1, -20, 150, -500, 625)
Four roots = (5.0, 5.0, 5.0, 5.0)
Equal roots detected: True
2 pairs of equal roots, B4 = 0.
Coefficients = (1, -6, -11, 60, 100)
Four roots = (5.0, -2.0, 5.0, -2.0)
Equal roots detected: True
2 complex roots.
Coefficients = (4, 4, -75, -776, -1869)
Four roots = [7.0, -3.0, (-2.5+4j), (-2.5-4j)]
Equal roots detected: False
2 complex roots, B4 = 0.
Coefficients = (1, -12, 33, 18, -208)
Four roots = ((3+2j), (3-2j), 8.0, -2.0)
Equal roots detected: False
4 complex roots.
Coefficients = (1, -20, 408, 2296, 18020)
Four roots = [(13+19j), (13-19j), (-3+5j), (-3-5j)]
Equal roots detected: False
4 complex roots, B4 = 0.
Coefficients = (1, -12, 83, -282, 442)
Four roots = ((3+5j), (3-5j), (3+2j), (3-2j))
Equal roots detected: False
2 pairs of equal complex roots, B4 = 0.
Coefficients = (1, -12, 62, -156, 169)
Four roots = ((3+2j), (3-2j), (3+2j), (3-2j))
Equal roots detected: TrueWhen description contains note depressed quartic was processed as quadratic in Template:RoundBoxBottom Template:RoundBoxBottom
Two Conic Sections
Examples of conic sections include: ellipse, circle, parabola and hyperbola.
This section presents examples of two conic sections, circle and ellipse, and how to calculate the coordinates of the point/s of intersection, if any, of the two sections.
Let one section with name have equation
Let other section with name have equation
Because there can be as many as 4 points of intersection, a special "resolvent" quartic function is used to calculate the coordinates of the point/s of intersection.
Coefficients of associated "resolvent" quartic are calculated as follows:
# python code
def intersection_of_2_conic_sections (abcdef, ABCDEF) :
'''
A_,B_,C_,D_,E_ = intersection_of_2_conic_sections (abcdef, ABCDEF)
where A_,B_,C_,D_,E_ are coefficients of associated resolvent quartic function:
y = f(x) = A_*x**4 + B_*x**3 + C_*x**2 + D_*x + E_
'''
A,B,C,D,E,F = ABCDEF
a,b,c,d,e,f = abcdef
G = ((-1)*(B)*(a) + (1)*(A)*(b))
H = ((-1)*(B)*(d) + (1)*(D)*(b))
I = ((-1)*(B)*(f) + (1)*(F)*(b))
J = ((-1)*(C)*(a) + (1)*(A)*(c))
K = ((-1)*(C)*(d) + (-1)*(E)*(a) + (1)*(A)*(e) + (1)*(D)*(c))
L = ((-1)*(C)*(f) + (-1)*(E)*(d) + (1)*(D)*(e) + (1)*(F)*(c))
M = ((-1)*(E)*(f) + (1)*(F)*(e))
g = ((-1)*(C)*(b) + (1)*(B)*(c))
h = ((-1)*(E)*(b) + (1)*(B)*(e))
i = ((-1)*(A)*(b) + (1)*(B)*(a))
j = ((-1)*(D)*(b) + (1)*(B)*(d))
k = ((-1)*(F)*(b) + (1)*(B)*(f))
A_ = ((-1)*(J)*(g) + (1)*(G)*(i))
B_ = ((-1)*(J)*(h) + (-1)*(K)*(g) + (1)*(G)*(j) + (1)*(H)*(i))
C_ = ((-1)*(K)*(h) + (-1)*(L)*(g) + (1)*(G)*(k) + (1)*(H)*(j) + (1)*(I)*(i))
D_ = ((-1)*(L)*(h) + (-1)*(M)*(g) + (1)*(H)*(k) + (1)*(I)*(j))
E_ = ((-1)*(M)*(h) + (1)*(I)*(k))
str1 = 'y = ({})x^4 + ({})x^3 + ({})x^2 + ({})x + ({}) '.format(A_,B_,C_,D_,E_)
print (str1)
return A_,B_,C_,D_,E_
With no common point
Resolvent quartic function (black curve) has no real roots.
axis of quartic function is compressed to illustrate shape of curve.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
has no real roots. Therefore, there is no point of intersection. Template:RoundBoxBottom
With one common point
Resolvent quartic function (black curve) has two equal, real roots.
axis of quartic function is compressed to illustrate shape of curve.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
Roots of are:
has 2 equal, real roots at effectively 1 real root where
Therefore, there is one point of intersection where Template:RoundBoxBottom
With two common points
Example 1
Resolvent quartic function (black curve) has two unique, real roots.
axis of quartic function is compressed to illustrate shape of curve.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
Roots of are:
has 2 unique, real roots at
Therefore, there are two points of intersections where Template:RoundBoxBottom
Example 2
Resolvent quartic function (black curve) has two pairs of equal roots.
axis of quartic function is compressed to illustrate shape of curve.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
Roots of are:
has 2 pairs of equal roots at effectively 2 real roots.
Therefore, there are two points of intersection where Template:RoundBoxBottom
With 3 common points
Resolvent quartic function (black curve) has one pair of equal roots and 2 unique, real roots.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
Roots of are:
has 1 pair of equal roots at and 2 unique, real roots at effectively 3 real roots.
Therefore, there are three points of intersection where Template:RoundBoxBottom
With 4 common points
Resolvent quartic function (black curve) has 4 unique, real roots.
axis of quartic function is compressed to illustrate shape of curve.
Let ellipse (red curve) have equation:
Let circle (blue curve) have equation:
Then, resolvent quartic function (black curve) has equation:
Roots of are:
has 4 real roots as shown above.
Therefore, there are four points of intersection where Template:RoundBoxBottom