To solve the equation [tex]\( 2v^2 - 3v - 26 = (v - 2)^2 \)[/tex] for [tex]\( v \)[/tex], let's follow these steps:
1. Expand the right-hand side: Start by expanding [tex]\((v - 2)^2\)[/tex].
[tex]\[
(v - 2)^2 = v^2 - 4v + 4
\][/tex]
Thus, the original equation becomes:
[tex]\[
2v^2 - 3v - 26 = v^2 - 4v + 4
\][/tex]
2. Move all terms to one side: To set up the equation for solving, bring all terms to one side of the equation so that the other side equals zero.
[tex]\[
2v^2 - 3v - 26 - (v^2 - 4v + 4) = 0
\][/tex]
3. Combine like terms: Simplify the left-hand side by combining like terms.
[tex]\[
2v^2 - 3v - 26 - v^2 + 4v - 4 = 0
\][/tex]
[tex]\[
(2v^2 - v^2) + (-3v + 4v) + (-26 - 4) = 0
\][/tex]
[tex]\[
v^2 + v - 30 = 0
\][/tex]
4. Solve the quadratic equation: We now have a standard quadratic equation [tex]\( v^2 + v - 30 = 0 \)[/tex]. To find the solutions for [tex]\( v \)[/tex], we can factor the quadratic equation.
We look for two numbers that multiply to [tex]\(-30\)[/tex] and add up to [tex]\(1\)[/tex]. These numbers are [tex]\(6\)[/tex] and [tex]\(-5\)[/tex].
[tex]\[
v^2 + v - 30 = (v + 6)(v - 5) = 0
\][/tex]
5. Set each factor to zero and solve for [tex]\( v \)[/tex]:
[tex]\[
v + 6 = 0 \quad \text{or} \quad v - 5 = 0
\][/tex]
Solving these equations gives:
[tex]\[
v = -6 \quad \text{or} \quad v = 5
\][/tex]
Therefore, the solutions to the equation [tex]\( 2v^2 - 3v - 26 = (v - 2)^2 \)[/tex] are [tex]\( v = -6 \)[/tex] and [tex]\( v = 5 \)[/tex].
