# How do you solve sqrt(x-8)+sqrt(x+3)=1?

Jan 30, 2016

$\sqrt{x - 8} + \sqrt{x + 3} = 1$ has no solutions

#### Explanation:

$\sqrt{x - 8} + \sqrt{x + 3} = 1$

$\implies {\left(\sqrt{x - 8} + \sqrt{x + 3}\right)}^{2} = {1}^{2}$

$\implies \left(x - 8\right) + 2 \sqrt{\left(x - 8\right) \left(x + 3\right)} + \left(x + 3\right) = 1$

$\implies 2 \sqrt{{x}^{2} - 5 x - 24} = - 2 x + 6$

$\implies \sqrt{{x}^{2} - 5 x - 24} = - x + 3$

$\implies {\left(\sqrt{{x}^{2} - 5 x - 24}\right)}^{2} = {\left(- x + 3\right)}^{2}$

$\implies {x}^{2} - 5 x - 24 = {x}^{2} - 6 x + 9$

$\implies x = 33$

In the process of squaring, we may have generated extraneous solutions, and so we must check to see if our possible solution is valid.

$\sqrt{33 - 8} + \sqrt{33 + 3} = \sqrt{25} + \sqrt{36} = 5 + 6 = 11$

As the only possible value for $x$ we found is not a solution, there is no solution. As another way of seeing this, note that the graph of
$\sqrt{x - 8} + \sqrt{x + 3} - 1$
never intersects the $x$ axis, and thus the equivalent equation
$\sqrt{x - 8} + \sqrt{x + 3} - 1 = 0$
has no solutions.

graph{sqrt(x-8)+sqrt(x+3)-1 [-3.92, 76.08, -4, 36]}