What is the positive difference between the two real solutions to the equation (x - 1)^2 =… — ACT

What is the positive difference between the two real solutions to the equation $(x - 1)^2 = 5x - 5$ ?

The positive difference between the two real solutions is 5.

Expanding the left side of

(x - 1)^2 = 5x - 5

yields

x^2 - 2x + 1 = 5x - 5

. Moving all terms to the left side by subtracting

5x

and adding

5

gives the standard quadratic equation

x^2 - 7x + 6 = 0

. Factoring this expression gives

(x - 1)(x - 6) = 0

, which yields the solutions

1

and

6

. The positive difference between these solutions is

6 - 1 = 5

Expand the squared binomial on the left side of the equation.

x^2 - 2x + 1 = 5x - 5

Before factoring a quadratic equation, all terms must be expanded and moved to one side to set the equation equal to zero.

Subtract

5x

and add

5

to both sides to rearrange the equation into standard quadratic form.

x^2 - 7x + 6 = 0

This sets the equation equal to zero, which is a prerequisite for using the zero product property.

Factor the quadratic expression by finding two numbers that multiply to

6

and add to

-7

(x - 1)(x - 6) = 0

Factoring allows us to split the quadratic equation into two linear equations.

Set each factor to zero to solve for

x

x = 1

and

x = 6

By the zero product property, if the product of two factors is zero, at least one of the factors must be zero.

Subtract the smaller solution from the larger solution to find the positive difference.

6 - 1 = 5

The question asks for the positive difference between the two solutions.

Solving quadratic equations by expanding, rearranging into standard form, and factoring over the integers.

Question