Question

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

Answer 1

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

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

Explanation:

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

Answer 2

Answer: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

Step-by-step explanation: