To solve for [tex]\( y \)[/tex] in the equation [tex]\(\sqrt{-4y + 21} = y\)[/tex], let's follow these steps:
1. Square both sides of the equation:
[tex]\[
\left(\sqrt{-4y + 21}\right)^2 = y^2
\][/tex]
This simplifies to:
[tex]\[
-4y + 21 = y^2
\][/tex]
2. Rearrange the equation to set it to zero:
[tex]\[
y^2 + 4y - 21 = 0
\][/tex]
3. Apply the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] to solve the quadratic equation [tex]\( y^2 + 4y - 21 = 0 \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -21 \)[/tex]:
[tex]\[
y = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-21)}}{2 \cdot 1}
\][/tex]
[tex]\[
y = \frac{-4 \pm \sqrt{16 + 84}}{2}
\][/tex]
[tex]\[
y = \frac{-4 \pm \sqrt{100}}{2}
\][/tex]
[tex]\[
y = \frac{-4 \pm 10}{2}
\][/tex]
4. Find the solutions by solving the two possible values for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-4 + 10}{2}
\][/tex]
[tex]\[
y = \frac{6}{2}
\][/tex]
[tex]\[
y = 3
\][/tex]
and
[tex]\[
y = \frac{-4 - 10}{2}
\][/tex]
[tex]\[
y = \frac{-14}{2}
\][/tex]
[tex]\[
y = -7
\][/tex]
5. Check the solutions in the original equation to ensure they are valid (since we squared the equation, additional solutions might have been introduced):
Check [tex]\( y = 3 \)[/tex]:
[tex]\[
\sqrt{-4(3) + 21} = 3
\][/tex]
[tex]\[
\sqrt{-12 + 21} = 3
\][/tex]
[tex]\[
\sqrt{9} = 3
\][/tex]
[tex]\[
3 = 3 \quad \text{(This is a valid solution)}
\][/tex]
Check [tex]\( y = -7 \)[/tex]:
[tex]\[
\sqrt{-4(-7) + 21} = -7
\][/tex]
[tex]\[
\sqrt{28 + 21} = -7
\][/tex]
[tex]\[
\sqrt{49} = -7
\][/tex]
[tex]\[
7 = -7 \quad \text{(This is not valid since 7 is not equal to -7)}
\][/tex]
Only [tex]\( y = 3 \)[/tex] is a valid solution to the equation.
Therefore, the solution is:
[tex]\[
y = 3
\][/tex]
