(x-2)^2-6(x-2)+5=0

2 min read Jun 17, 2024

Solving the Quadratic Equation: (x-2)^2 - 6(x-2) + 5 = 0

This article will guide you through solving the quadratic equation (x-2)^2 - 6(x-2) + 5 = 0.

Understanding the Equation

The equation is presented in a form that suggests a substitution could simplify it. Notice the repeated term (x-2). This is a clear indicator that we can use substitution to make the equation easier to work with.

Substitution

Let's substitute y = (x-2). This transforms our original equation into:

y^2 - 6y + 5 = 0

This is now a standard quadratic equation in terms of y.

Solving the Quadratic Equation

We can now solve for y using the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / 2a

Where:

a = 1
b = -6
c = 5

Substituting these values into the quadratic formula:

y = (6 ± √((-6)^2 - 4 * 1 * 5)) / (2 * 1)

y = (6 ± √(16)) / 2

y = (6 ± 4) / 2

This gives us two possible solutions for y:

y1 = 5
y2 = 1

Back Substitution

Now we need to substitute back x-2 for y to find the solutions for x:

x - 2 = 5 => x = 7
x - 2 = 1 => x = 3

Solutions

Therefore, the solutions to the quadratic equation (x-2)^2 - 6(x-2) + 5 = 0 are x = 7 and x = 3.