Which system of equations can you

use to find the roots of the equation

2x3 + 4x2 - x + 5 = - 3x2 + 4x + 9?

y = 2x3 + x2 + 3x + 5

y = 9

y = 2x3 + x2

y = 3x + 14

y = 2x3 + 4x2 - x + 5

y = - 3x2 + 4x + 9

The answer is y = 2x3 + 4x2 - x + 5 and y = - 3x2 + 4x + 9

Roots are both: x=-4, x = - 1/2, x = 1

Proof:

Solve for x over the real numbers:

2 x^3 + 4 x^2 - x + 5 = - 3 x^2 + 4 x + 9

Subtract - 3 x^2 + 4 x + 9 from both sides:

2 x^3 + 7 x^2 - 5 x - 4 = 0

The left hand side factors into a product with three terms:

(x - 1) (x + 4) (2 x + 1) = 0

Split into three equations:

x - 1 = 0 or x + 4 = 0 or 2 x + 1 = 0

Add 1 to both sides:

x = 1 or x + 4 = 0 or 2 x + 1 = 0

Subtract 4 from both sides:

x = 1 or x = - 4 or 2 x + 1 = 0

Subtract 1 from both sides:

x = 1 or x = - 4 or 2 x = - 1

Divide both sides by 2:

Answer: x = 1 or x = - 4 or x = - 1/2