Cubic Curves (function)
iax3 + bx2 + cx + d
ALL cubic equations follow the base ax3 + bx2 + cx + d. When solved, the equations end up with 3 roots. The roots and 0s affect the slope and placement of the parabolas when graphed.
Examples of the equation include:
3+x2−16x=16
−n3−n2+22n+40=0
y(y2+2y)=19y+20
k3+9k2+26k+24=0
x3+2x2−50=25x
−p3+19p=30
6x2−x3=5x+12
The function had been originally published by Gerolamo Cardano (1501-1576) but, as hinted by Niccola Tartaglia, Cardano was not the original discoverer of this function.
Cubic functions all share this property. They all have either 1, or 3 real roots.
ALL cubic equations follow the base ax3 + bx2 + cx + d. When solved, the equations end up with 3 roots. The roots and 0s affect the slope and placement of the parabolas when graphed.
Examples of the equation include:
3+x2−16x=16
−n3−n2+22n+40=0
y(y2+2y)=19y+20
k3+9k2+26k+24=0
x3+2x2−50=25x
−p3+19p=30
6x2−x3=5x+12
The function had been originally published by Gerolamo Cardano (1501-1576) but, as hinted by Niccola Tartaglia, Cardano was not the original discoverer of this function.
Cubic functions all share this property. They all have either 1, or 3 real roots.
The use of the cubic function is not very common, although it has still be used in several different scenarios including the curves of fonts such as Georgia and Times New Roman. Cubic curves have also been used in Kepler’s heliocentric law of planetary motion and the finding of eigenvalues in 3x3 matrixes.
Normally the way to SOLVE the cubic function is to simplify it into a quadratic and then solve that. Here's a guide on how to solve a simple cubic equation (x^3 + 3x^2 - 6x - 18)
Step 1:
Group the cubic into two groups. This will make it easier to solve.
( x^3 + 3x^2 ) and ( -6x - 18 )
Step 2:
Factorise the grouped equations by one of their factors.
*Common factor of x^3 and 3x^2 would be x^2, and common factor of -6x - 18 would be -6
You should come up with something along the lines of x^2(x+3) - -6(x+3) = 0
Step 3:
Because each of the terms share the same factor (which is x+3), you can group the factors together.
This turns into (x^2 - 6)(x+3)
Step 4:
As x^2 - 6 turns into x - the root of 6, and x + the root of 6 this means you can create the equation
(x - root of 6) (x + root of 6) (x+3)
If you look at the roots, you can find the solution.
The answers are 3, root of 6, and negative root of 6.
Normally the way to SOLVE the cubic function is to simplify it into a quadratic and then solve that. Here's a guide on how to solve a simple cubic equation (x^3 + 3x^2 - 6x - 18)
Step 1:
Group the cubic into two groups. This will make it easier to solve.
( x^3 + 3x^2 ) and ( -6x - 18 )
Step 2:
Factorise the grouped equations by one of their factors.
*Common factor of x^3 and 3x^2 would be x^2, and common factor of -6x - 18 would be -6
You should come up with something along the lines of x^2(x+3) - -6(x+3) = 0
Step 3:
Because each of the terms share the same factor (which is x+3), you can group the factors together.
This turns into (x^2 - 6)(x+3)
Step 4:
As x^2 - 6 turns into x - the root of 6, and x + the root of 6 this means you can create the equation
(x - root of 6) (x + root of 6) (x+3)
If you look at the roots, you can find the solution.
The answers are 3, root of 6, and negative root of 6.