Multiply both sides by sqrt(2kv+k).

Question

Multiply both sides by sqrt(2kv+k).

Asked on April 24, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-24T15:08:13.542Z

You're on a roll — To make $k$ the subject of the formula $r = \frac{2v+4}{\sqrt{2kv+k}}$ and simplify, we need to isolate $k$. Step 1: Multiply both sides by $\sqrt{2kv+k}$. $$ r\sqrt{2kv+k} = 2v+4 $$ Step 2: Square both sides to eliminate the square root. $$ (r\sqrt{2kv+k})^2 = (2v+4)^2 $$ $$ r^2(2kv+k) = (2v+4)^2 $$ Step 3: Expand the left side and factor out $k$. $$ 2kvr^2 + kr^2 = (2v+4)^2 $$ $$ k(2vr^2 + r^2) = (2v+4)^2 $$ Step 4: Isolate $k$ by dividing both sides by $(2vr^2 + r^2)$. $$ k = \frac{(2v+4)^2}{2vr^2 + r^2} $$ Step 5: Simplify the denominator by factoring out $r^2$. $$ k = \frac{(2v+4)^2}{r^2(2v+1)} $$ The expression for $k$ is: $$ \boxed{k = \frac{(2v+4)^2}{r^2(2v+1)}} $$ 4 down, 1 left today. What's next?

Accepted Answer · 2026-04-24T15:08:13.542Z

You're on a roll — To make $k$ the subject of the formula $r = \frac{2v+4}{\sqrt{2kv+k}}$ and simplify, we need to isolate $k$. Step 1: Multiply both sides by $\sqrt{2kv+k}$. $$ r\sqrt{2kv+k} = 2v+4 $$ Step 2: Square both sides to eliminate the square root. $$ (r\sqrt{2kv+k})^2 = (2v+4)^2 $$ $$ r^2(2kv+k) = (2v+4)^2 $$ Step 3: Expand the left side and factor out $k$. $$ 2kvr^2 + kr^2 = (2v+4)^2 $$ $$ k(2vr^2 + r^2) = (2v+4)^2 $$ Step 4: Isolate $k$ by dividing both sides by $(2vr^2 + r^2)$. $$ k = \frac{(2v+4)^2}{2vr^2 + r^2} $$ Step 5: Simplify the denominator by factoring out $r^2$. $$ k = \frac{(2v+4)^2}{r^2(2v+1)} $$ The expression for $k$ is: $$ \boxed{k = \frac{(2v+4)^2}{r^2(2v+1)}} $$ 4 down, 1 left today. What's next?

Multiply both sides by sqrt(2kv+k).