If x and y are positive real numbers such that x^2 - y^2 = 105 and x + y = 15, what is the… — GMAT

If $x$ and $y$ are positive real numbers such that $x^2 - y^2 = 105$ and $x + y = 15$ , what is the value of $(x - 2y)^2$ ?

Answer: 9

The value of

(x - 2y)^2

is 9.

Factoring

x^2 - y^2

into

(x - y)(x + y)

gives

(x - y)(15) = 105

, which simplifies to

x - y = 7

. Solving the system of equations

x + y = 15

and

x - y = 7

gives

x = 11

and

y = 4

. Substituting these values into the target expression

(x - 2y)^2

yields

(11 - 2(4))^2 = 3^2 = 9

Apply the difference of squares factoring identity to

x^2 - y^2

(x - y)(x + y) = 105

The difference of two squares

x^2 - y^2

factors into

(x - y)(x + y)

Calculate the value of

x - y

x - y = 7

Since

x + y = 15

, dividing 105 by 15 gives

x - y = 7

Solve the system of equations for

x

and

y

x = 11

and

y = 4

Adding

(x + y) + (x - y) = 15 + 7

yields

2x = 22 \implies x = 11

. Subtracting

(x + y) - (x - y) = 15 - 7

yields

2y = 8 \implies y = 4

Evaluate the expression

(x - 2y)^2

9

Substitute

x = 11

and

y = 4

into

(x - 2y)^2

to obtain

(11 - 2(4))^2 = (11 - 8)^2 = 3^2 = 9

Difference of Squares Factoring and Systems of Equations

Question