# How do you solve sqrt (100 - d^2) = 10 - d?

We have $10 - d \ge 0$ or $10 \ge d$

Hence

$\sqrt{100 - {d}^{2}} = 10 - d$

$\sqrt{\left(10 + d\right) \left(10 - d\right)} = \sqrt{{\left(10 - d\right)}^{2}}$

For $d \ne 10$ we have that

$\sqrt{\frac{10 + d}{10 - d}} = 1$

Take squares in both sides

$\frac{10 + d}{10 - d} = 1$

$10 + d = 10 - d$

$d = 0$

Hence the solutions are $d = 0$ and $d = 10$

Jun 1, 2018

$d = 0$ and $10$

#### Explanation:

$\sqrt{100 - {d}^{2}} = 10 - d$

First, square both sides:
${\left(\sqrt{100 - {d}^{2}}\right)}^{2} = {\left(10 - d\right)}^{2}$

$100 - {d}^{2} = 100 - 20 d + {d}^{2}$

Subtract $\textcolor{b l u e}{100}$ from both sides of the equation:
$100 - {d}^{2} \quad \textcolor{b l u e}{- \quad 100} = 100 - 20 d + {d}^{2} \quad \textcolor{b l u e}{- \quad 100}$

$- {d}^{2} = - 20 d + {d}^{2}$

Add $\textcolor{b l u e}{{d}^{2}}$ to both sides of the equation:
$- {d}^{2} \quad \textcolor{b l u e}{+ \quad {d}^{2}} = - 20 d + {d}^{2} \quad \textcolor{b l u e}{+ \quad {d}^{2}}$

$0 = 2 {d}^{2} - 20 d$

Factor out a $\textcolor{b l u e}{2 d}$:
$0 = 2 d \left(d - 10\right)$

$2 d = 0$ and $d - 10 = 0$

$d = 0$ and $d = 10$

$- - - - - - - - - - - - - - - - - - -$

Now plug in both solutions to make sure they are really solutions:

First plug in $0$:
$\sqrt{100 - {d}^{2}} = 10 - d$

$\sqrt{100 - 0} = 10 - 0$

$\sqrt{100} = 10$

$10 = 10$

This is true. Therefore, $0$ is a solution.

Now plug in $10$:
$\sqrt{100 - {d}^{2}} = 10 - d$

$\sqrt{100 - {10}^{2}} = 10 - 10$

$\sqrt{100 - 100} = 0$

$\sqrt{0} = 0$

$0 = 0$

This is also a solution.

Hope this helps!