(x-2)^2-4x+8=0

3 min read Jun 17, 2024

Solving the Quadratic Equation: (x-2)^2 - 4x + 8 = 0

This article will walk through the steps of solving the quadratic equation (x-2)^2 - 4x + 8 = 0. We will explore different methods to find the solutions, also known as roots, of this equation.

Expanding and Simplifying

First, let's simplify the equation by expanding the squared term:

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

Substituting this back into the original equation, we get:

x^2 - 4x + 4 - 4x + 8 = 0

Combining like terms:

x^2 - 8x + 12 = 0

Solving by Factoring

Now we have a simplified quadratic equation. One way to solve it is by factoring. We need to find two numbers that multiply to 12 and add up to -8. These numbers are -6 and -2:

x^2 - 6x - 2x + 12 = 0

Factoring by grouping:

x(x - 6) - 2(x - 6) = 0

(x - 6)(x - 2) = 0

Therefore, the solutions are:

x = 6 and x = 2

Solving by Quadratic Formula

Another method to solve the equation is using the quadratic formula:

x = [-b ± √(b^2 - 4ac)] / 2a

Where a = 1, b = -8, and c = 12.

Substituting these values into the formula:

x = [8 ± √((-8)^2 - 4 * 1 * 12)] / (2 * 1)

x = [8 ± √(64 - 48)] / 2

x = [8 ± √16] / 2

x = [8 ± 4] / 2

This gives us two solutions:

x = (8 + 4) / 2 = 6

x = (8 - 4) / 2 = 2

Conclusion

We have successfully solved the quadratic equation (x-2)^2 - 4x + 8 = 0 using both factoring and the quadratic formula. Both methods yielded the same solutions: x = 6 and x = 2.

It's important to note that the quadratic formula can be used to solve any quadratic equation, even those that are not easily factored.