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I am trying to multiply and simplify the following radical expression.

$$(\sqrt{x}+5 - 4)(\sqrt{x}+5+4)$$

According to the book, the answer is $x - 11$

However, I am confused about how this even works. I tried using the following calculator that shows all the steps. http://www.softmath.com/math-com-calculator/adding-matrices/multiply-radical-expressions.html#c=simplify_algstepssimplify&v217=%2528%25u221Ax%2B5%2520-%25204%2529%2528%2520%25u221Ax%2B5%2B4%2529

However, the answer is completely different when using the calculator $x + 10\sqrt{x} + 9$.

I know I must be missing something here and it is probably something simple. I can simplify radicals by themselves with no problem, but when they are multiplied that is when I get into trouble.

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    $\begingroup$ The book's answer corresponds to this product $$\left(\sqrt{x+5}-4\right)\left(\sqrt{x+5}+4\right)$$ but you entered $$\left(\sqrt{x}+5-4\right)\left(\sqrt{x}+5+4\right)$$ into the calculator. See the difference? $\endgroup$ – Blue Feb 17 at 15:11
  • $\begingroup$ Thanks, I knew I was missing something simple. $\endgroup$ – Curt Rand Feb 17 at 15:17
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You have: $$(\sqrt{x+5}-4)(\sqrt{x+5}+4)$$ Your lack of collecting the $+5$ under the square root is why your calculator procured the incorrect answer.

Instead, use Difference of Two Squares here: $$(A-B)(A+B)=A^2-B^2$$ (achievable by expanding)

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  • $\begingroup$ I knew I was missing something stupidly simple. Thank you for your help. $\endgroup$ – Curt Rand Feb 17 at 15:17
  • $\begingroup$ No problem. Always have to be careful when displaying math in calculators. $\endgroup$ – Rhys Hughes Feb 17 at 15:21

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