To solve the equation [tex]\( \sqrt{12v - 27} = v \)[/tex], let's proceed step-by-step.
1. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{12v - 27})^2 = v^2
\][/tex]
This simplifies to:
[tex]\[
12v - 27 = v^2
\][/tex]
2. Rearrange the equation to standard quadratic form:
[tex]\[
v^2 - 12v + 27 = 0
\][/tex]
3. Solve the quadratic equation using the quadratic formula:
The quadratic formula is given by:
[tex]\[
v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\( v^2 - 12v + 27 = 0 \)[/tex], the coefficients are:
[tex]\[
a = 1, \quad b = -12, \quad c = 27
\][/tex]
Substituting these into the quadratic formula we get:
[tex]\[
v = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 1 \cdot 27}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
v = \frac{12 \pm \sqrt{144 - 108}}{2}
\][/tex]
[tex]\[
v = \frac{12 \pm \sqrt{36}}{2}
\][/tex]
[tex]\[
v = \frac{12 \pm 6}{2}
\][/tex]
4. Calculate the two possible solutions:
[tex]\[
v_1 = \frac{12 + 6}{2} = \frac{18}{2} = 9
\][/tex]
[tex]\[
v_2 = \frac{12 - 6}{2} = \frac{6}{2} = 3
\][/tex]
Therefore, the solutions to the equation [tex]\( \sqrt{12v - 27} = v \)[/tex] are:
[tex]\[
v = 3, 9
\][/tex]
