If x is a positive real number such that \sqrt{x} - \frac{2}{\sqrt{x}} = 3, what is the va… — GMAT

If $x$ is a positive real number such that $\sqrt{x} - \frac{2}{\sqrt{x}} = 3$ , what is the value of $x + \frac{4}{x}$ ?

The value of

x + \frac{4}{x}

is 13.

Squaring both sides of

\sqrt{x} - \frac{2}{\sqrt{x}} = 3

gives

(\sqrt{x})^2 - 2(\sqrt{x})(\frac{2}{\sqrt{x}}) + (\frac{2}{\sqrt{x}})^2 = 9

, which simplifies to

x - 4 + \frac{4}{x} = 9

. Adding 4 to both sides yields

x + \frac{4}{x} = 13

. Thus, the option with value 13 is correct.

Square both sides of the given equation

\sqrt{x} - \frac{2}{\sqrt{x}} = 3

{\left(\sqrt{x} - \frac{2}{\sqrt{x}}\right)}^2 = 3^2

Squaring both sides relates the terms

\sqrt{x}

and

\frac{2}{\sqrt{x}}

x

and

\frac{4}{x}

Expand the left side using the perfect square identity

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

(\sqrt{x})^2 - 2\left(\sqrt{x}\right)\left(\frac{2}{\sqrt{x}}\right) + \left(\frac{2}{\sqrt{x}}\right)^2 = 9

Applying the algebraic identity expands the squared expression correctly.

Simplify the terms on the left side.

x - 4 + \frac{4}{x} = 9

The product

\sqrt{x} \cdot \frac{2}{\sqrt{x}}

simplifies to

2

, making the middle term

-2 \cdot 2 = -4

Add 4 to both sides to isolate

x + \frac{4}{x}

x + \frac{4}{x} = 13

Isolating the target algebraic expression yields the final value.

Squaring binomial radical expressions using the identity

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

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