Problem 29 - Algebra 2

Question

Solve each equation for the given variable. $$ \frac{1}{c}-\frac{c}{a^{2}-b^{2}}=0 ; c $$

Step-by-Step Solution

Answer

The value of \(c\) is \(\sqrt{a^2 - b^2}\)

Step 1: Write down the equation

The given equation is: \( \frac{1}{c}-\frac{c}{a^2-b^2} = 0 \)

Step 2: Add \(\frac{c}{a^{2}-b^{2}}\) to both sides of the equation

This gives the equation: \( \frac{1}{c} = \frac{c}{a^{2}-b^{2}} \)

Step 3: Cross-multiply

Cross-multiplication gives: \( c^2 = a^2 - b^2 \)

Step 4: Find the value of \(c\).

Since \(c\) must be positive (as it is in the denominator of the initial expression), \(c\) is the positive square root of \(a^2 - b^2\). That is \(c = \sqrt{a^2 - b^2}\).