Find the value of x if $\frac{\sqrt{2}}{x + \sqrt{2}}$ = $\frac{1}{x - \sqrt{2}}$

Analysis

To solve for $x$ in the equation $\frac{\sqrt{2}}{x + \sqrt{2}} = \frac{1}{x - \sqrt{2}}$, we need to eliminate the fractions by cross-multiplying and then isolate the variable $x$.

Concept

This is a linear equation involving surds. The key principle here is cross-multiplication: if $\frac{a}{b} = \frac{c}{d}$, then $ad = bc$. Once the fractions are removed, we group the terms containing $x$ on one side and the constant terms on the other.

Step-by-Step Solution

Step 1: Cross-multiply the equation.

Step 2: Expand the brackets. Multiply $\sqrt{2}$ by each term inside the parenthesis on the left side:

(Note: $\sqrt{2} \times \sqrt{2} = 2$)

Step 3: Group the terms with $x$ on one side. Subtract $x$ from both sides and add $2$ to both sides:

Step 4: Factor out $x$ on the left side.

Step 5: Solve for $x$ by dividing both sides by $(\sqrt{2} - 1)$.

Step 6: Rationalize the denominator. To simplify, multiply the numerator and denominator by the conjugate of the denominator, which is $(\sqrt{2} + 1)$:

Expand the numerator:

Expand the denominator (difference of two squares):

So, $x = \frac{4 + 3\sqrt{2}}{1} = 3\sqrt{2} + 4$.

Conclusion

Comparing our result $3\sqrt{2} + 4$ with the given options, it matches option A.

Exam Tip

When you see a denominator with a surd (like $\sqrt{2} - 1$), always remember to rationalize it by multiplying by its conjugate (change the sign in the middle). This is a common requirement in JAMB Mathematics to simplify your final answer into the format provided in the options.