If x and y are non-zero real numbers such that 9x^2 - 12xy + 4y^2 = 0, what is the value o… — GRE General Test

If $x$ and $y$ are non-zero real numbers such that $9x^2 - 12xy + 4y^2 = 0$ , what is the value of $\frac{3x + y}{2y}$ ?

The value of the expression is

\frac{3}{2}

The given expression

9x^2 - 12xy + 4y^2

factors into the perfect square

(3x - 2y)^2 = 0

. Setting the base equal to zero gives

3x = 2y

. Substituting

2y

in place of

3x

in the target expression yields

\frac{2y + y}{2y} = \frac{3y}{2y} = \frac{3}{2}

Factor the quadratic expression

9x^2 - 12xy + 4y^2 = (3x - 2y)^2 = 0

The equation is a perfect square trinomial of the form

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

where

a = 3x

and

b = 2y

Solve for the relationship between

x

and

y

3x - 2y = 0 \implies 3x = 2y

Taking the square root of both sides gives a linear relationship between

3x

and

2y

Substitute

3x = 2y

into the targeted expression

\frac{3x + y}{2y} = \frac{2y + y}{2y} = \frac{3y}{2y} = \frac{3}{2}

Replacing

3x

with

2y

eliminates

x

and simplifies the expression directly.

Perfect Square Trinomial Factoring

Question