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I'm just stuck on a problem where I have to simplify an expression. This is the expression:

$\sqrt{x^2\:-\:\left(\frac{x}{2}\right)^2}$

The textbook has the answer as:

$\frac{\sqrt{3}x}{2}$

I have no idea how to get to that. I've tried to figure it out for the past hour and I have tried online solvers but I can't understand what the steps are. Any help would be very much appreciated.

Thanks!

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  • $\begingroup$ What have you tried so far? $\endgroup$ – perpetuallyconfused Jan 12 at 2:21
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    $\begingroup$ $x^2-\left(\dfrac x2\right)^2=\dfrac{3x^2}4$ $\endgroup$ – J. W. Tanner Jan 12 at 2:28
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    $\begingroup$ A little bit picky, but the text-book is wrong. The expression actually simplifies to $\frac {|x|\sqrt3}{2}$ $\endgroup$ – Doug M Jan 12 at 2:36
  • $\begingroup$ @DougM Not picky at all, I would say. $\endgroup$ – saulspatz Jan 12 at 2:40
  • $\begingroup$ J.W. I understand that but where do you get the 3 from? $\endgroup$ – Scratch Cat Jan 12 at 2:49
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One step a time!

You solution assumes that $x\ge 0$.

\begin{align} x^2-(\frac{x}{2})^2 &=x^2-\frac{x^2}{4}\\ &=1\cdot x^2-\frac{1}{4}\cdot x^2\\ &=(1-\frac{1}{4})x^2= \frac34 x^2 \end{align}

So $$ \sqrt{x^2-(\frac{x}{2})^2}=\sqrt{\frac34 x^2} =\frac{\sqrt{3}}{\sqrt{4}}\sqrt{x^2}=\frac{\sqrt{3}}{2}x\quad x\ge 0 $$

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