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

April 24, 2026Mathematics

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Answer

k = \frac{(2v+4)^2}{r^2(2v+1)}

5 steps

To make $k$ the subject of the formula $r = \frac{2v+4}{\sqrt{2kv+k}}$ and simplify, we need to isolate $k$ .

Multiply both sides by $\sqrt{2kv+k}$ .
$r\sqrt{2kv+k} = 2v+4$
Square both sides to eliminate the square root.
$(r\sqrt{2kv+k})^2 = (2v+4)^2$ $r^2(2kv+k) = (2v+4)^2$
Expand the left side and factor out $k$ .
$2kvr^2 + kr^2 = (2v+4)^2$ $k(2vr^2 + r^2) = (2v+4)^2$
Isolate $k$ by dividing both sides by $(2vr^2 + r^2)$ .
$k = \frac{(2v+4)^2}{2vr^2 + r^2}$
Simplify the denominator by factoring out $r^2$ .
$k = \frac{(2v+4)^2}{r^2(2v+1)}$
The expression for $k$ is:

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