To solve the equation
[tex]\[
2v^2 + 20v + 42 = (v + 7)^2
\][/tex]
we will follow these steps:
1. Expand the Right Side of the Equation:
[tex]\[
(v + 7)^2 = v^2 + 14v + 49
\][/tex]
So the equation becomes:
[tex]\[
2v^2 + 20v + 42 = v^2 + 14v + 49
\][/tex]
2. Move All Terms to One Side of the Equation to Set It to Zero:
Subtract [tex]\(v^2 + 14v + 49\)[/tex] from both sides:
[tex]\[
2v^2 + 20v + 42 - (v^2 + 14v + 49) = 0
\][/tex]
Simplify the left side:
[tex]\[
2v^2 + 20v + 42 - v^2 - 14v - 49 = 0
\][/tex]
Combine like terms:
[tex]\[
v^2 + 6v - 7 = 0
\][/tex]
3. Factor the Quadratic Equation:
We need to factor [tex]\(v^2 + 6v - 7 = 0\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-7\)[/tex] and add to [tex]\(6\)[/tex]. These numbers are [tex]\(7\)[/tex] and [tex]\(-1\)[/tex]. So, we factor the quadratic as:
[tex]\[
(v + 7)(v - 1) = 0
\][/tex]
4. Solve for [tex]\(v\)[/tex]:
Set each factor equal to zero:
[tex]\[
v + 7 = 0 \quad \text{or} \quad v - 1 = 0
\][/tex]
Solving these equations gives:
[tex]\[
v = -7 \quad \text{or} \quad v = 1
\][/tex]
So, the solutions are:
[tex]\[
v = -7, 1
\][/tex]
