Question

Solve the equation by completing square method.

$v^2+6v=40$

Short answer

The solutions are$v=-10,v=4$.

Step-by-step solution

Given Information.  

The Given equation is$v^2+6v=40$.

Isolate the variable terms on one side and the constant terms on the other.

The variable terms are on the left side.

$v^2+6v=40$

The coefficient of $v$is $6$.

So,$b=6$

Find 12b2and add to the both sides of equation.

$$\begin{gathered} ={\left(\frac{1}{2}\left(6\right)\right)}^2 \\ ={\left(3\right)}^2=9 \end{gathered}$$

Add $9$both sides.

$$\begin{gathered} v^2+6v+9=40+9 \\ {v}^2+6v+9=49 \end{gathered}$$

Factor the perfect square trinomial as a binomial square.

$(v+3{)}^2=49$

Use the square root property.

$$\begin{gathered} v+3=\pm \sqrt{49} \\ v+3=\pm 7 \\ v=-3\pm 7 \end{gathered}$$

Rewrite to show two solutions.

$$\begin{gathered} v=-3+7=4 \\ v=-3-7=-10 \\ v=-10,v=4 \end{gathered}$$

Check the solutions.

Substitute $v=-10$

$$\begin{gathered} v^2+6v=40 \\ (-10{)}^2+6(-10)\stackrel{?}{=}40 \\ 100-60\stackrel{?}{=}40 \\ 40=40True \end{gathered}$$

Substitute $v=4$

$$\begin{gathered} v^2+6v=40 \\ {\left(4\right)}^2+6\left(4\right)\stackrel{?}{=}40 \\ 16+24\stackrel{?}{=}40 \\ 40=40 \end{gathered}$$