If x and y are non-zero real numbers such that x^2 - 5xy + 6y^2 = 0, what is the sum of al… — GMAT

If $x$ and $y$ are non-zero real numbers such that $x^2 - 5xy + 6y^2 = 0$ , what is the sum of all possible values of $\frac{x^2 + y^2}{xy}$ ?

The sum of all possible values of

\frac{x^2 + y^2}{xy}

\frac{35}{6}

Factoring

x^2 - 5xy + 6y^2 = 0

yields

(x - 2y)(x - 3y) = 0

. Thus,

\frac{x}{y} = 2

\frac{x}{y} = 3

. Simplifying

\frac{x^2 + y^2}{xy}

gives

\frac{x}{y} + \frac{y}{x}

. Substituting

2

yields

2 + \frac{1}{2} = \frac{5}{2}

, and substituting

3

yields

3 + \frac{1}{3} = \frac{10}{3}

. The sum of these values is

\frac{5}{2} + \frac{10}{3} = \frac{35}{6}

Factor the quadratic equation

x^2 - 5xy + 6y^2 = 0

in terms of

x

and

y

(x - 2y)(x - 3y) = 0

Factoring determines the proportional relationships between

x

and

y

Solve for the possible ratio values of

\frac{x}{y}

Either

x = 2y \implies \frac{x}{y} = 2

, or

x = 3y \implies \frac{x}{y} = 3

Since

y \neq 0

, dividing by

y

gives the linear root ratios.

Rewrite the target expression

\frac{x^2 + y^2}{xy}

as a sum of ratios.

\frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x}

Splitting the numerator simplifies substitution of

\frac{x}{y}

Evaluate the expression for each possible ratio.

Case 1 (

x/y = 2

2 + \frac{1}{2} = \frac{5}{2}

. Case 2 (

x/y = 3

3 + \frac{1}{3} = \frac{10}{3}

Each root ratio produces a distinct value for the target expression.

Sum the two values obtained.

\frac{5}{2} + \frac{10}{3} = \frac{15 + 20}{6} = \frac{35}{6}

The question asks for the sum of all possible values of the expression.

Homogeneous Quadratic Equations and Algebraic Factoring

Estimated Time:2m 0s

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