Because the function is "quartic" (maximum power of ${\displaystyle x}$ is ${\displaystyle 4}$), the function contains exactly ${\displaystyle 4}$ 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.

Graph of typical quartic function showing minima and maximum.

The figure shows a typical quartic function.

The function crosses the ${\displaystyle X}$ axis in 4 different places. The function has 4 roots: ${\displaystyle (-2,0),(1,0),(5,0),(10,0).}$

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(${\displaystyle x}$) is very large, ${\displaystyle f(x)}$ is always positive.

• If absolute minimum is above ${\displaystyle X}$ axis, curve does not cross ${\displaystyle X}$ axis and function contains only complex roots.

• There is always at least one point where the curve is parallel to ${\displaystyle X}$ axis.

The curve is never parallel to the ${\displaystyle Y}$ axis. For any real value of ${\displaystyle x}$ there is always a real value of ${\displaystyle y.}$

When coefficient d is missing, there is a stationary point at x = 0.

If coefficient ${\displaystyle d}$ is missing, the quartic function becomes ${\displaystyle y=ax^{4}+bx^{3}+cx^{2}+e,}$ and

${\displaystyle y'=4ax^{3}+3bx^{2}+2cx=x(4ax^{2}+3bx+2c).}$

For a stationary point ${\displaystyle y'=x(4ax^{2}+3bx+2c)=0.}$

When coefficient ${\displaystyle d}$ is missing, there is always a stationary point at ${\displaystyle x=0.}$

Graph of quartic function that is quadratic in ${\displaystyle x^{2}}$.
Because coefficient ${\displaystyle d}$ is missing, there is a stationary point where ${\displaystyle x=0.}$

If coefficients ${\displaystyle b,d}$ are missing, the quartic function becomes a quadratic in ${\displaystyle x^{2}.}$

The curve (red line) in diagram has equation: ${\displaystyle y=f(x)={\frac {x^{4}-13x^{2}+36}{5}}}$

The quartic equation may be solved as: ${\displaystyle X^{2}-13X+36=0}$ where ${\displaystyle X=x^{2}}$ or ${\displaystyle x={\sqrt {X}}.}$

${\displaystyle X=4}$ or ${\displaystyle X=9.}$

${\displaystyle x=\pm 2}$ or ${\displaystyle x=\pm 3.}$

The quartic function may be expressed as ${\displaystyle x=ay^{4}+by^{3}+cy^{2}+dy+e.}$

Unless otherwise noted, references to "quartic function" on this page refer to function of form ${\displaystyle y=ax^{4}+bx^{3}+cx^{2}+dx+e.}$

Graph of quartic function with coefficient a negative.
There is no absolute minimum.

Coefficient ${\displaystyle a}$ may be negative as shown in diagram.

As abs${\displaystyle (x)}$ increases, the value of ${\displaystyle f(x)}$ is dominated by the term ${\displaystyle -ax^{4}.}$

When abs${\displaystyle (x)}$ is very large, ${\displaystyle f(x)}$ is always negative.

Unless stated otherwise, any reference to "quartic function" on this page will assume coefficient ${\displaystyle a}$ positive.

Graph of quartic function with coefficient b missing.
Sum of roots is ${\displaystyle 0.}$
${\displaystyle Y}$axis compressed for clarity.

When sum of roots is ${\displaystyle 0,}$ coefficient ${\displaystyle b=0.}$

In the diagram, roots of ${\displaystyle f(x)}$ are ${\displaystyle -5,-4,2,7.}$

Sum of roots ${\displaystyle =0.}$

Therefore coefficient ${\displaystyle b=0.}$

Graphs of quartic function and associated cubic function.

When ${\displaystyle p}$ is a root of the function, the function may be expressed as:

${\displaystyle (x-p)(Ax^{3}+Bx^{2}+Cx+D)}$ where

${\displaystyle A=a;\ B=Ap+b;\ C=Bp+c;\ D=Cp+d.}$

When one real root ${\displaystyle p}$ is known, the other three roots may be calculated as roots of the cubic function ${\displaystyle Ax^{3}+Bx^{2}+Cx+D.}$

In the diagram the quartic function has equation: ${\displaystyle y={\frac {x^{4}-23x^{3}+163x^{2}-393x+252}{48}}.}$

It is known that ${\displaystyle 3}$ is a root of this function.

The associated cubic has equation: ${\displaystyle y={\frac {x^{3}-20x^{2}+103x-84}{48}}}$

The 2 curves coincide at points ${\displaystyle (1,0),\ (7,0),\ (12,0),}$ the three points that are roots of both functions. ${\displaystyle }$

Figure 1. Quartic 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:

${\displaystyle (-5,0),(-2,0),(1,0),(3,-6),(6,2)}$

Rearrange the standard quartic function to prepare for the calculation of ${\displaystyle a,b,c,d,e:}$

${\displaystyle x^{4}a+x^{3}b+x^{2}c+xd+1e-y=0.}$

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)

Quartic function defined by the 5 points ${\displaystyle (-5,0),(-2,0),(1,0),(3,-6),(6,2)}$ is ${\displaystyle y={\frac {0.875x^{4}+0.15x^{3}-27.975x^{2}-24.05x+51}{33}}.}$

Figure 1. Quartic 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 ${\displaystyle (-2,-2)\ =}$ slope at ${\displaystyle (6,-4)\ =\ 0.}$

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:

${\displaystyle (-2,-2),(6,-4),(4,1)}$

At point ${\displaystyle (-2,-2)}$ slope is ${\displaystyle 0.}$

At point ${\displaystyle (6,-4)}$ slope is ${\displaystyle 0.}$

Rearrange the standard quartic function to prepare for the calculation of ${\displaystyle a,b,c,d,e:}$

${\displaystyle x^{4}a+x^{3}b+x^{2}c+xd+1e-y=0.}$

Rearrange the standard cubic function of slope to prepare for the calculation of ${\displaystyle a,b,c,d,e:}$

${\displaystyle 4x^{3}a+3x^{2}b+2xc+1d+0e-s=0.}$

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)

Quartic function defined by three points and two slopes is: ${\displaystyle y={\frac {1.5625x^{4}-12.125x^{3}-14.75x^{2}+136.5x+114.0}{48}}.}$

Associated cubic functions

When p == -2 Figure 1. Quartic function and associated cubic function when p = -2. In this case roots of associated cubic include x = p. Quartic function is: ${\displaystyle y=f(x)={\frac {1.5625x^{4}-12.125x^{3}-14.75x^{2}+136.5x+114.0}{48}}.}$ When ${\displaystyle p==-2,}$ associated cubic function is : ${\displaystyle y=g(x)={\frac {1.5625x^{3}-15.25x^{2}+15.75x+105}{48}}.}$ Three blue vertical lines show 3 values of ${\displaystyle x}$ where ${\displaystyle g(x)=0}$ and ${\displaystyle f(x)=f(-2)}$ In this case roots of ${\displaystyle g(x)}$ include ${\displaystyle x=p.}$ When p == 5 Figure 1. Quartic function and associated cubic function when p = 5. In this case the one root of associated cubic excludes x = p. Quartic function is: ${\displaystyle y=f(x)={\frac {1.5625x^{4}-12.125x^{3}-14.75x^{2}+136.5x+114.0}{48}}.}$ When ${\displaystyle p==5,}$ associated cubic function is : ${\displaystyle y=g(x)={\frac {1.5625x^{3}-4.3125x^{2}-36.3125x-45.0625}{48}}.}$ Two blue vertical lines show 2 values of ${\displaystyle x}$ where ${\displaystyle f(x)=f(5)}$ In this case the one root of ${\displaystyle g(x)}$ excludes ${\displaystyle x=p.}$ When p == 6 Figure 1. Quartic function and associated cubic function when p = 6. In this case the one root of associated cubic includes x = p. Quartic function is: ${\displaystyle y=f(x)}$${\displaystyle ={\frac {1.5625x^{4}-12.125x^{3}-14.75x^{2}+136.5x+114.0}{48}}.}$ When ${\displaystyle p==6,}$ associated cubic function is: ${\displaystyle y=g(x)}$${\displaystyle ={\frac {1.5625x^{3}-2.75x^{2}-31.25x-51}{48}}.}$ One blue vertical line shows 1 value of ${\displaystyle x}$ where ${\displaystyle f(x)=f(6)}$ In this case the one root of ${\displaystyle g(x)}$ includes ${\displaystyle x=p.}$

Graphs of quartic function and first two derivatives.
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: ${\displaystyle y=f(x)={\frac {x^{4}-16x^{3}+42x^{2}+12x+4}{144}}.}$

The first derivative, the red line, has equation: ${\displaystyle y'=g(x)={\frac {x^{3}-12x^{2}+21x+3}{36}}.}$

The second derivative, the blue line, has equation: ${\displaystyle y''=h(x)={\frac {x^{2}-8x+7}{12}}.}$

When ${\displaystyle x<x_{1}:}$

• ${\displaystyle y'}$ is increasing.

• ${\displaystyle y''}$ is positive.

• ${\displaystyle f(x)}$ is always concave up.

When ${\displaystyle x==x_{1}:}$

• ${\displaystyle y'}$ is at a local maximum.

• ${\displaystyle y''=0.}$

• Concavity of ${\displaystyle f(x)}$ is between up and down.

When ${\displaystyle x_{1}<x<x_{2}:}$

• ${\displaystyle y'}$ is decreasing.

• ${\displaystyle y''}$ is negative.

• ${\displaystyle f(x)}$ is always concave down.

When ${\displaystyle x==x_{2}:}$

• ${\displaystyle y'}$ is at a local minimum.

• ${\displaystyle y''=0.}$

• Concavity of ${\displaystyle f(x)}$ is between down and up.

When ${\displaystyle x_{2}<x:}$

• ${\displaystyle y'}$ is increasing.

• ${\displaystyle y''}$ is positive.

• ${\displaystyle f(x)}$ is always concave up.

Graphs of quartic function and first two derivatives showing maximum and minima.
Point ${\displaystyle p_{2}}$ on ${\displaystyle f(x)}$ is a stationary point. ${\displaystyle f(x)}$ at point ${\displaystyle p_{2}}$ is concave down. Point ${\displaystyle p_{2}}$ is local maximum.
Point ${\displaystyle p_{1}}$ on ${\displaystyle f(x)}$ is a stationary point. ${\displaystyle f(x)}$ at point ${\displaystyle p_{1}}$ is concave up. Point ${\displaystyle p_{1}}$ is local minimum.
Similarly, point ${\displaystyle p_{3}}$ is local minimum.

In the diagram the black line has equation: ${\displaystyle y=f(x)={\frac {1.5625x^{4}-12.125x^{3}-14.75x^{2}+136.5x+114}{48}}.}$

The first derivative, the red line, has equation: ${\displaystyle y'=g(x)={\frac {6.25x^{3}-36.375x^{2}-29.5x+136.5}{48}}.}$

The second derivative, the blue line, has equation: ${\displaystyle y''=h(x)={\frac {18.75x^{2}-72.75x-29.5}{48}}.}$

Roots of ${\displaystyle g(x):\ x_{1}=-2;\ x_{2}=1.82;\ x_{3}=6.}$

Let point ${\displaystyle p_{1}}$ on ${\displaystyle f(x)}$ have coordinates ${\displaystyle (x_{1},f(x_{1})).}$

At ${\displaystyle x_{1}\ h(x_{1})}$ is positive. Point ${\displaystyle p_{1}}$ is a stationary point and ${\displaystyle f(x)}$ at ${\displaystyle p_{1}}$ is concave up. Point ${\displaystyle p_{1}}$ is a local minimum.

Let point ${\displaystyle p_{2}}$ on ${\displaystyle f(x)}$ have coordinates ${\displaystyle (x_{2},f(x_{2})).}$

At ${\displaystyle x_{2}\ h(x_{2})}$ is negative. Point ${\displaystyle p_{2}}$ is a stationary point and ${\displaystyle f(x)}$ at ${\displaystyle p_{2}}$ is concave down. Point ${\displaystyle p_{2}}$ is a local maximum.

Let point ${\displaystyle p_{3}}$ on ${\displaystyle f(x)}$ have coordinates ${\displaystyle (x_{3},f(x_{3})).}$

At ${\displaystyle x_{3}\ h(x_{3})}$ is positive. Point ${\displaystyle p_{3}}$ is a stationary point and ${\displaystyle f(x)}$ at ${\displaystyle p_{3}}$ is concave up. Point ${\displaystyle p_{3}}$ is a local minimum.

Graph of quartic function with 2 stationary points and first 2 derivatives.
Black line: ${\displaystyle y=f(x)=4(x^{4}+3x^{3}+3x^{2}+x)-1.}$
Red line: ${\displaystyle y'=g(x)=4(4x^{3}+9x^{2}+6x+1).}$
${\displaystyle y''=h(x)=4(12x^{2}+18x+6)}$
Blue line: ${\displaystyle y''=4(2x^{2}+3x+1)={\frac {h(x)}{6}}.}$
Dotted portion of black line shows where ${\displaystyle f(x)}$ is concave down.

In the diagram, point ${\displaystyle p_{1}}$ on ${\displaystyle f(x)}$ has coordinates ${\displaystyle (x_{1},f(x_{1})).}$

Similarly, points ${\displaystyle p_{2},p_{3}}$ have coordinates ${\displaystyle (x_{2},f(x_{2})),\ (x_{3},f(x_{3})).}$

${\displaystyle y'}$ has roots: ${\displaystyle x_{1}=-1;\ x_{3}=-0.25.}$

Points ${\displaystyle p_{1},p_{3}}$ are stationary points.

${\displaystyle y''}$ has roots: ${\displaystyle x_{1}=-1;\ x_{2}=-0.5.}$

Points ${\displaystyle p_{1},p_{2}}$ are points of inflection.

At point ${\displaystyle p_{3}\ y''}$ is positive. ${\displaystyle f(x)}$ at ${\displaystyle p_{3}}$ is concave up. Point ${\displaystyle p_{3}}$ is local minimum.

Graph of quartic function with no equal roots.

Red line in diagram is of function: ${\displaystyle y=f(x)={\frac {x^{4}-3x^{3}-x^{2}-27x-90}{50}}}$

${\displaystyle a,b,c,d,e=1,-3,-1,-27,-90}$

${\displaystyle R,S=15269148,-35977608}$

${\displaystyle x_{1}={\frac {-S}{R}}={\frac {35977608}{15269148}}=2.3562289133617\dots }$

${\displaystyle r,s=35977608,-60634332}$

${\displaystyle x_{2}={\frac {-s}{r}}={\frac {60634332}{35977608}}=1.685335278543253\dots }$

${\displaystyle x_{1}!=x_{2}.}$ There are no equal roots.

Graph of quartic function with exactly 2 equal roots.

Red line in diagram is of function: ${\displaystyle y=f(x)={\frac {x^{4}+6x^{3}-48x^{2}-182x+735}{100}}}$

${\displaystyle a,b,c,d,e=1,6,-48,-182,735}$

${\displaystyle R,S=-1027353600,-7191475200}$

${\displaystyle x_{1}={\frac {-S}{R}}={\frac {7191475200}{-1027353600}}=-7}$

${\displaystyle r,s=7191475200,50340326400}$

${\displaystyle x_{2}={\frac {-s}{r}}={\frac {-50340326400}{7191475200}}=-7}$

${\displaystyle x_{1}=x_{2}=-7.}$ There are 2 equal roots at ${\displaystyle x=-7.}$

# 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)

Graph of quartic function with exactly 3 equal roots and corresponding quadratic.

Red line in diagram is of function: ${\displaystyle y=f(x)={\frac {x^{4}-2x^{3}-36x^{2}+162x-189}{100}}}$

${\displaystyle a,b,c,d,e=1,-2,-36,162,-189}$

${\displaystyle R,S=0,0\ \dots \dots \ (1c)}$

${\displaystyle r,s=0,0\ \dots \dots \ (2c)}$

In this case the calculation of ${\displaystyle x_{1},x_{2}}$ is not appropriate because there are more than 2 equal roots. Try equations ${\displaystyle (1b),(2b).}$ Both of these are equivalent to: ${\displaystyle y=g(x)=x^{2}-6x+9,}$ blue line in diagram.

Discriminant of ${\displaystyle g(x)=(-6)^{2}-4(1)(9)=0.\ g(x)}$ has two equal roots at ${\displaystyle x={\frac {-(-6)}{2(1)}}=3.}$ Therefore ${\displaystyle f(x)}$ has 3 equal roots at ${\displaystyle x=3.}$

Graph of quartic function with 4 equal roots and corresponding cubic.
${\displaystyle g(x)={\frac {f'(x)}{4}}.}$

Red line in diagram is of function: ${\displaystyle y=f(x)=x^{4}-20x^{3}+150x^{2}-500x+625.}$

${\displaystyle a,b,c,d,e=1,-20,150,-500,625}$

${\displaystyle R,S=0,0}$

${\displaystyle r,s=0,0}$

${\displaystyle K,L,M=0,0,0}$

${\displaystyle k,l,m=0,0,0}$

In this case ${\displaystyle (1b),(2b),(1c),(2c)}$ are all null.

This is the only case in which ${\displaystyle (1b),(2b)}$ are null.

${\displaystyle (1a),(2a)}$ are both equivalent to: ${\displaystyle y=g(x)=x^{3}-15x^{2}+75x-125,}$ blue line in diagram.

${\displaystyle g(x)}$ has one root at ${\displaystyle x=5.}$ Therefore ${\displaystyle f(x)}$ has 4 equal roots at ${\displaystyle x=5.}$