If x is a real number satisfying the equation 2x^2 - 7x + 3 = 0 and y is a real number sat… — GMAT

If $x$ is a real number satisfying the equation $2x^2 - 7x + 3 = 0$ and $y$ is a real number satisfying the equation $y^2 + 5y - 14 = 0$ , what is the maximum possible value of $\frac{x}{y}$ ?

$\frac{3}{2}$ Answer
B
$\frac{3}{7}$
C
$\frac{1}{4}$
D
$-\frac{3}{7}$
E
$6$

Answer

The maximum possible value of

\frac{x}{y}

\frac{3}{2}

Factoring

2x^2 - 7x + 3 = 0

gives

(2x - 1)(x - 3) = 0

, so

x \in \left\{\frac{1}{2}, 3\right\}

. Factoring

y^2 + 5y - 14 = 0

gives

(y + 7)(y - 2) = 0

, so

y \in \{-7, 2\}

. The four possible values for

\frac{x}{y}

are

\frac{3}{2}

-\frac{3}{7}

\frac{1}{4}

, and

-\frac{1}{14}

. The greatest value among these is

\frac{3}{2}

Step-by-Step Solution

Solve the quadratic equation for

x

by factoring.

Factor

2x^2 - 7x + 3 = 0

(2x - 1)(x - 3) = 0

, yielding solutions

x = \frac{1}{2}

and

x = 3

Finding all valid real roots of the first equation determines possible values for the numerator.

Solve the quadratic equation for

y

by factoring.

Factor

y^2 + 5y - 14 = 0

(y + 7)(y - 2) = 0

, yielding solutions

y = -7

and

y = 2

Finding all valid real roots of the second equation determines possible values for the denominator.

Evaluate all possible combinations for the ratio

\frac{x}{y}

The possible ratios are

\frac{3}{2} = 1.5

\frac{3}{-7} = -\frac{3}{7}

\frac{1/2}{2} = \frac{1}{4} = 0.25

, and

\frac{1/2}{-7} = -\frac{1}{14}

Testing all root pairs ensures we identify the maximum overall value.

Select the maximum value among the evaluated ratios.

The largest value is

\frac{3}{2}

Comparing

1.5

-0.428

0.25

, and

-0.071

shows

1.5

is the greatest.

Key Concept

Solving quadratic equations via polynomial factoring and optimizing rational expressions over discrete solution sets.

Estimated Time:1m 30s

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