What is the sum of all real solutions to the equation (x - 3)^2 = 4(x - 3)? — GMAT

What is the sum of all real solutions to the equation $(x - 3)^2 = 4(x - 3)$ ?

The sum of all real solutions to the equation is 10.

To find all solutions to

(x - 3)^2 = 4(x - 3)

, bring all terms to the left side:

(x - 3)^2 - 4(x - 3) = 0

. Factoring out

(x - 3)

yields

(x - 3)(x - 7) = 0

. This gives two distinct real roots,

x = 3

and

x = 7

. Adding these together yields a sum of 10.

Rearrange the given equation so that all terms are on one side.

(x - 3)^2 - 4(x - 3) = 0

Moving all terms to one side allows for factoring without losing variable solutions.

Factor out the common algebraic factor

(x - 3)

(x - 3)\,[(x - 3) - 4] = 0 \implies (x - 3)(x - 7) = 0

Factoring avoids the mistake of dividing by a variable expression.

Set each factor equal to zero to find the real roots.

x - 3 = 0 \implies x = 3

and

x - 7 = 0 \implies x = 7

A product of factors equals zero if and only if at least one factor is zero.

Calculate the sum of the real solutions.

3 + 7 = 10

The question asks for the sum of all distinct real solutions.

Factoring Quadratic Expressions and Avoiding Variable Cancellation

Estimated Time:1m 0s

Question