Q42. - Algebra 1

Question

Solve each equation. Check your solution.

$\sqrt{3x-14}+x=6$

Step-by-Step Solution

Answer

The solution of the given equation is $x=12$.

 

Check:

When $x=12$,

Substitute 12 for x in the given equation $\sqrt{x+4}=x-8$.

width="110" height="115" style="max-width: none; vertical-align: -58px;" $$\begin{gathered} \sqrt{x+4}=x-8 \\ \sqrt{12+4}\underset{¯}{\underset{¯}{?}}12-8 \\ \sqrt{16}\underset{¯}{\underset{¯}{?}}4 \\ \text{ 4}=4 \end{gathered}$$

Therefore, $x=12$ is the solution of the given equation $\sqrt{x+4}=x-8$.

 

When $x=5$,

Substitute 5 for x in the given equation $\sqrt{x+4}=x-8$.

$$\begin{gathered} \sqrt{x+4}=x-8 \\ \sqrt{5+4}\underset{¯}{\underset{¯}{?}}5-8 \\ \sqrt{9}\underset{¯}{\underset{¯}{?}}-3 \\ \text{ 3}\ne -3 \end{gathered}$$

Therefore, $x=5$ is not the solution of the given equation $\sqrt{x+4}=x-8$.

Step 1. Rewrite the given equation.

The given equation can be written as:

$$\begin{gathered} \sqrt{3x-14}+x=6 \\ \sqrt{3x-14}=6-x \\ {\left(\sqrt{3x-14}\right)}^2={\left(6-x\right)}^2\left(\text{squaring both sides}\right) \\ 3x-14={\left(6-x\right)}^2 \\ 3x-14=6^2-2\left(6\right)\left(x\right)+x^2\left(\because {\left(a-b\right)}^2=a^2-2ab+b^2\right) \\ 3x-14=36-12x+x^2 \end{gathered}$$

Step 2. Solve the given equation.

The solution of the given equation is:

$$\begin{gathered} 3x-14=36-12x+x^2 \\ {x}^2-12x-3x+36+14=0 \\ x^2-15x+50=0 \\ x^2-10x-5x+50=0 \\ x\left(x-10\right)-5\left(x-10\right)=0 \\ \left(x-10\right)\left(x-5\right)=0 \\ x-10=0\text{ or }x-5=0 \\ x=10\text{ or }x=5 \end{gathered}$$

Therefore, the values x are 10 and 5.

Step 3. Check the solution.

When $x=10$,

Substitute 10 for x in the given equation $\sqrt{3x-14}+x=6$.

$$\begin{gathered} \sqrt{3x-14}+x=6 \\ \sqrt{3\left(10\right)-14}+10\underset{¯}{\underset{¯}{?}}6 \\ \sqrt{30-14}+10\underset{¯}{\underset{¯}{?}}6 \\ \sqrt{16}+10\underset{¯}{\underset{¯}{?}}6 \\ 4+10\underset{¯}{\underset{¯}{?}}6 \\ \text{ 14}\ne 6 \end{gathered}$$

Therefore, $x=10$ is not the solution of the given equation $\sqrt{3x-14}+x=6$.

 

When $x=5$,

Substitute 5 for x in the given equation $\sqrt{3x-14}+x=6$.

$$\begin{gathered} \sqrt{3x-14}+x=6 \\ \sqrt{3\left(5\right)-14}+5\underset{¯}{\underset{¯}{?}}6 \\ \sqrt{15-14}+5\underset{¯}{\underset{¯}{?}}6 \\ \sqrt{1}+5\underset{¯}{\underset{¯}{?}}6 \\ \text{ 1}+5\underset{¯}{\underset{¯}{?}}6 \\ \text{ 6}=6 \end{gathered}$$

Therefore, $x=5$ is the solution of the given equation $\sqrt{3x-14}+x=6$.