Question

In the following exercises, solve:

$\sqrt{12c+6}=\sqrt{10-4c}$.

Short answer

The solution is,$c=\frac{1}{4}$.

Step-by-step solution

Step 1. Given Information.

The expression is,

$\sqrt{12c+6}=\sqrt{10-4c}$.

Step 2. Finding the solution.

$\sqrt{12c+6}=\sqrt{10-4c}$

Squaring both sides,

role="math" localid="1647933105428" $$\begin{gathered} \Rightarrow {\left(\sqrt{12c+6}\right)}^2={\left(\sqrt{10-4c}\right)}^2 \\ \Rightarrow 12c+6=10-4c \end{gathered}$$

Subtracting $6$from both sides,

$$\begin{gathered} \Rightarrow 12c=10-4c-6 \\ \Rightarrow 12c=-4c+4 \end{gathered}$$

Adding $4c$on both sides,

$$\begin{gathered} \Rightarrow 12c+4c=4 \\ \Rightarrow 16c=4 \end{gathered}$$

Dividing both sides by role="math" localid="1647933260020" $16$,

role="math" localid="1647933248751" $$\begin{gathered} \Rightarrow \frac{16c}{16}=\frac{4}{16} \\ \Rightarrow c=\frac{1}{4} \end{gathered}$$

Step 3. Checking the solution.

$\sqrt{12c+6}=\sqrt{12\times \frac{1}{4}+6}$

$$\begin{gathered} =\sqrt{3+6} \\ =\sqrt{9} \end{gathered}$$

$\sqrt{10-4c}=\sqrt{10-4\times \frac{1}{4}}$

$$\begin{gathered} =\sqrt{10-1} \\ =\sqrt{9} \end{gathered}$$

The solution is true.