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Can this problem be simplified further?

$\sqrt{(x+5)(x-7)}$

My initial thought was to set one side to zero and the square both sides but so I have been stuck. Its been way to long since I've had to do this and do not remember how to get rid of the radical sign.

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2 Answers

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What you are asking about is the simplification of an expression $$\sqrt{(x+5)(x-7)}$$

There is no equation to solve here, so you can not pretend that it equals zero, and then solve for $x$. That means, you need to check whether $(x+5)(x-7)$ is perhaps a square.

But, note that $$(x + 5)(x - 7) = x^2 - 2x - 35 = (x^2 - 2x + 1) - 36 = (x - 1)^2 - 36$$ and hence, $(x+5)(x-7)$ is not a perfect square of the form $y^2$, so you're stuck with the radical remaining.

Hence, $$\sqrt{(x+5)(x-7)}$$ is as simplified as it gets.

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This depends on the assumptions on the domain for $x$ and $y$. For real numbers, there is no simplification $(x+5)(x+7)=y^2$ in terms of $x$. However, should you be interested in finite fields, then over the field $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$ we have $$ (x+5)(x+7)=(x+1)^2, $$ so its square root is simply $x+1$.

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