Answer :

We start with the equation

[tex]$$10v^2 - 8 = 181.$$[/tex]

Step 1: Isolate the term with [tex]$v^2$[/tex].

Add [tex]$8$[/tex] to both sides of the equation:

[tex]$$10v^2 = 181 + 8 = 189.$$[/tex]

Step 2: Solve for [tex]$v^2$[/tex].

Divide both sides of the equation by [tex]$10$[/tex]:

[tex]$$v^2 = \frac{189}{10} = 18.9.$$[/tex]

Step 3: Solve for [tex]$v$[/tex].

Take the square root of both sides. Remember that taking the square root gives two solutions (one positive and one negative):

[tex]$$v = \sqrt{18.9} \quad \text{or} \quad v = -\sqrt{18.9}.$$[/tex]

Using a calculator, we find

[tex]$$\sqrt{18.9} \approx 4.347413023856832,$$[/tex]

so the two solutions are

[tex]$$v \approx 4.347413023856832 \qquad \text{and} \qquad v \approx -4.347413023856832.$$[/tex]