(x+4)(x-1)=-x^2+3x+4

3 min read Jun 16, 2024

Solving the Equation (x+4)(x-1) = -x^2 + 3x + 4

This article will guide you through the process of solving the equation (x+4)(x-1) = -x^2 + 3x + 4.

Step 1: Expanding the Left Side

The left side of the equation is a product of two binomials. We can expand it using the FOIL method (First, Outer, Inner, Last).

First: (x) * (x) = x^2
Outer: (x) * (-1) = -x
Inner: (4) * (x) = 4x
Last: (4) * (-1) = -4

Combining these terms, we get: (x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4

Step 2: Rewriting the Equation

Now our equation becomes: x^2 + 3x - 4 = -x^2 + 3x + 4

Step 3: Bringing All Terms to One Side

To solve for x, we need to bring all terms to one side of the equation. We can do this by adding x^2, subtracting 3x, and subtracting 4 from both sides.

2x^2 - 8 = 0

Step 4: Solving for x

The resulting equation is a quadratic equation. We can solve it by factoring or using the quadratic formula.

Factoring:

2x^2 - 8 = 0
2(x^2 - 4) = 0
2(x+2)(x-2) = 0
Therefore, x = -2 or x = 2

Quadratic Formula:

For the equation ax^2 + bx + c = 0, the quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 2, b = 0, and c = -8. Substituting these values into the quadratic formula, we get: x = (0 ± √(0^2 - 4 * 2 * -8)) / (2 * 2)
x = (± √64) / 4
x = (± 8) / 4
Therefore, x = 2 or x = -2

Conclusion

Therefore, the solutions to the equation (x+4)(x-1) = -x^2 + 3x + 4 are x = 2 and x = -2.