(x-5)(x+1)+5=0

2 min read Jun 17, 2024

Solving the Quadratic Equation: (x-5)(x+1) + 5 = 0

This article will guide you through solving the quadratic equation (x-5)(x+1) + 5 = 0. We'll break down the steps and explain the concepts involved.

Expanding the Equation

First, we need to expand the equation by multiplying out the factors:

(x-5)(x+1) + 5 = 0

FOIL Method:
- First: x * x = x²
- Outer: x * 1 = x
- Inner: -5 * x = -5x
- Last: -5 * 1 = -5

This gives us: x² - 4x - 5 + 5 = 0

Simplifying the Equation

Now, we can simplify the equation by combining like terms:

x² - 4x = 0

Solving for x

This is now a simple quadratic equation. We can solve for x by factoring:

Factor out x: x(x - 4) = 0

To satisfy the equation, either x = 0 or (x-4) = 0.

Therefore, the solutions are:
- x = 0
- x = 4

Verification

We can verify our answers by plugging them back into the original equation:

For x = 0: (0 - 5)(0 + 1) + 5 = (-5)(1) + 5 = 0 (verified)
For x = 4: (4 - 5)(4 + 1) + 5 = (-1)(5) + 5 = 0 (verified)

Conclusion

We have successfully solved the quadratic equation (x-5)(x+1) + 5 = 0, finding that the solutions are x = 0 and x = 4.