(x-1)^2-5(x-1)^2+6=0

2 min read Jun 17, 2024

Solving the Quadratic Equation: (x-1)^2 - 5(x-1) + 6 = 0

This article will guide you through solving the quadratic equation (x-1)^2 - 5(x-1) + 6 = 0. We will utilize the factoring method to find the solutions for x.

Understanding the Equation

The equation is a quadratic equation in disguise. To make it more obvious, let's use a substitution:

Let y = (x - 1). Now, our equation becomes:

y^2 - 5y + 6 = 0

Factoring the Equation

We can now factor this quadratic equation:

Find two numbers that add up to -5 (the coefficient of y) and multiply to 6 (the constant term).
The numbers -2 and -3 satisfy these conditions.
Therefore, we can factor the equation as follows:

(y - 2)(y - 3) = 0

Solving for y

For the product of two terms to be zero, at least one of the terms must be zero. This gives us two possibilities:

y - 2 = 0
y - 3 = 0

Solving for y in each case:

y = 2
y = 3

Finding the Values of x

Now, we need to substitute back our original expression for y:

x - 1 = 2
x - 1 = 3

Solving for x in each case:

x = 3
x = 4

Conclusion

Therefore, the solutions to the quadratic equation (x-1)^2 - 5(x-1) + 6 = 0 are x = 3 and x = 4.