How do you solve sqrt(x-4)+ sqrt( x+1)=5?

Aug 24, 2016

$x = 8$

Explanation:

Lets just try something. You never know, it may work!

The standard move to get rid of square root is to square everything.

Write as :$\text{ } \sqrt{x - 4} = 5 - \sqrt{x + 1}$

Squaring both sides

$x - 4 = {5}^{2} - 10 \sqrt{x + 1} + \left(x + 1\right)$

$x - \left(x + 1\right) + 10 \sqrt{x + 1} = 25 + 4$

$- 1 + 10 \sqrt{x + 1} = 29$

$\sqrt{x + 1} = \frac{30}{10} = 3$

Square both sides

$x + 1 = 9$

$x = 8$

$\textcolor{b r o w n}{\text{Sometimes it is a matter of trying something. If that does not work}}$$\textcolor{b r o w n}{\text{then try something else.}}$

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check

$\textcolor{b l u e}{\sqrt{x - 4} + \sqrt{x + 1} = 5} \textcolor{g r e e n}{\text{ "->" } \sqrt{8 - 4} + \sqrt{8 + 1} = 5}$

$\implies \sqrt{4} + \sqrt{9} = 5$

$2 + 3 = 5 \leftarrow \text{ Left hand side=Right hand side so true}$