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 +5y =9y = 2x3 + x2 y = 3x + 14y = 2x3 + 4x2 – x + 5 y = –3x2 + 4x + 9
Accepted Solution
A:
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