Q27. - Algebra 1

Question

Solve each equation. Check your solution.

$\sqrt{2x-1}=2x-7$

Step-by-Step Solution

Answer

The solution of the given equation is $x=5$.

Step 1. Rewrite the given equation.

The given equation can be written as:

$$\begin{gathered} \sqrt{2x-1}=2x-7 \\ {\left(\sqrt{2x-1}\right)}^2={\left(2x-7\right)}^2\left(\text{squaring both sides}\right) \\ 2x-1={\left(2x\right)}^2-2\left(2x\right)\left(7\right)+{\left(7\right)}^2\left(\because {\left(a-b\right)}^2=a^2-2ab+b^2\right) \\ 2x-1=4x^2-28x+49 \end{gathered}$$

Step 2. Solve the given equation.

The solution of the given equation is:

$$\begin{gathered} 2x-1=4x^2-28x+49 \\ 4x^2-28x-2x+49+1=0 \\ 4x^2-30x+50=0 \\ 2\left(2x^2-15x+25\right)=0 \\ 2x^2-15x+25=0 \\ 2x^2-10x-5x+25=0 \\ 2x\left(x-5\right)-5\left(x-5\right)=0 \\ \left(x-5\right)\left(2x-5\right)=0 \\ x-5=0\text{ or }2x-5=0 \\ x=5\text{ or }x=\frac{5}{2} \end{gathered}$$

Therefore, the values of x are $x=5$ or $x=\frac{5}{2}$.

Step 3. Check the solutions.

Check the solution $x=5$,

Substitute 5 for x in the given equation $\sqrt{2x-1}=2x-7$.

$$\begin{gathered} \sqrt{2x-1}=2x-7 \\ \sqrt{2\left(5\right)-1}\underset{¯}{\underset{¯}{?}}2\left(5\right)-7 \\ \sqrt{10-1}\underset{¯}{\underset{¯}{?}}10-7 \\ \sqrt{9}\underset{¯}{\underset{¯}{?}}3 \\ \text{ 3}=3 \end{gathered}$$

Therefore, $x=5$ is the solution of the given equation $\sqrt{2x-1}=2x-7$.

 

Check the solution $x=\frac{5}{2}$,

Substitute $\frac{5}{2}$ for x in the given equation $\sqrt{2x-1}=2x-7$.

$$\begin{gathered} \sqrt{2x-1}=2x-7 \\ \sqrt{2\left(\frac{5}{2}\right)-1}\underset{¯}{\underset{¯}{?}}2\left(\frac{5}{2}\right)-7 \\ \sqrt{5-1}\underset{¯}{\underset{¯}{?}}5-7 \\ \sqrt{4}\underset{¯}{\underset{¯}{?}}-2 \\ \text{ 2}\ne -2 \end{gathered}$$

Therefore, $x=\frac{5}{2}$ is not the solution of the given equation $\sqrt{2x-1}=2x-7$.

Therefore, the only solution of the given equation $\sqrt{2x-1}=2x-7$ is $x=5$.