Given that \frac{\sqrt{x + 2} + \sqrt{x - 2}}{\sqrt{x + 2} - \sqrt{x - 2}} = 3, what is th… — JAMB UTME

Given that $\frac{\sqrt{x + 2} + \sqrt{x - 2}}{\sqrt{x + 2} - \sqrt{x - 2}} = 3$ , what is the value of $x$ ?

The value of

x

\frac{10}{3}

The correct solution is found by cross-multiplying the equation to get

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

. Grouping similar surd terms gives

4\sqrt{x - 2} = 2\sqrt{x + 2}

, which simplifies to

2\sqrt{x - 2} = \sqrt{x + 2}

. Squaring both sides yields

4(x - 2) = x + 2

, simplifying to

4x - 8 = x + 2

, which gives

3x = 10

and

x = \frac{10}{3}

Cross-multiply to clear the denominator

\sqrt{x + 2} + \sqrt{x - 2} = 3(\sqrt{x + 2} - \sqrt{x - 2})

Clear the fraction to group like radical terms on opposite sides.

Rearrange and combine like terms

4\sqrt{x - 2} = 2\sqrt{x + 2}

, which simplifies to

2\sqrt{x - 2} = \sqrt{x + 2}

Isolate the radical expressions.

Square both sides of the equation

(2\sqrt{x - 2})^2 = (\sqrt{x + 2})^2 \implies 4(x - 2) = x + 2

Eliminate radicals by squaring both sides, ensuring the coefficient

2

is squared to

4

Solve the linear equation for

x

4x - 8 = x + 2 \implies 3x = 10 \implies x = \frac{10}{3}

Isolate

x

to find the final value.

Solving Radical and Surd Equations

Question