What is the sum of the solutions to the equation 3(x - 2)^2 = -6x + 36? — ACT

What is the sum of the solutions to the equation $3(x - 2)^2 = -6x + 36$ ?

The sum of the solutions is 2.

The correct answer is 2. Expanding the equation correctly and rearranging it into standard form yields

x^2 - 2x - 8 = 0

. Factoring this expression gives

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

, which results in the solutions

x = 4

and

x = -2

. The sum of these two solutions is

4 + (-2) = 2

Divide both sides of the equation by 3 to simplify.

(x - 2)^2 = -2x + 12

Dividing both sides by the common factor of 3 simplifies the coefficients, making the algebraic manipulation and factoring steps easier.

Expand the squared binomial on the left side.

x^2 - 4x + 4 = -2x + 12

Expanding the binomial

(x - 2)^2

using the identity

(a - b)^2 = a^2 - 2ab + b^2

is necessary to write the equation in standard form.

Move all terms to the left side to set the equation equal to zero.

x^2 - 2x - 8 = 0

A quadratic equation must be in the standard form

ax^2 + bx + c = 0

to solve it by factoring.

Factor the quadratic trinomial.

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

We find two numbers that multiply to

-8

and add to

-2

. These numbers are

-4

and

2

Set each factor to zero to solve for the roots.

x = 4

x = -2

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

Calculate the sum of the solutions.

4 + (-2) = 2

The question asks for the sum of the solutions, so we add the two calculated values of

x

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

Question