To solve the equation [tex]\(\frac{11y - 2}{3} = y^2 + 2\)[/tex], we need to eliminate the fraction and solve for [tex]\(y\)[/tex]. Follow these steps:
1. Eliminate the fraction: Multiply both sides of the equation by 3 to clear the denominator.
[tex]\[
3 \cdot \frac{11y - 2}{3} = 3 \cdot (y^2 + 2)
\][/tex]
This simplifies to:
[tex]\[
11y - 2 = 3y^2 + 6
\][/tex]
2. Rearrange the equation: Move all terms to one side to set the equation to zero.
[tex]\[
11y - 2 - 3y^2 - 6 = 0
\][/tex]
Simplifying this, we get:
[tex]\[
-3y^2 + 11y - 8 = 0
\][/tex]
or equivalently:
[tex]\[
3y^2 - 11y + 8 = 0
\][/tex]
3. Solve the quadratic equation: We use the quadratic formula [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex] where [tex]\( a = 3 \)[/tex], [tex]\( b = -11 \)[/tex], and [tex]\( c = 8 \)[/tex].
Calculate the discriminant:
[tex]\[
b^2 - 4ac = (-11)^2 - 4(3)(8)
\][/tex]
[tex]\[
= 121 - 96
\][/tex]
[tex]\[
= 25
\][/tex]
Since the discriminant is positive, we have two distinct real solutions. Now, find the roots:
[tex]\[
y = \frac{-(-11) \pm \sqrt{25}}{2 \cdot 3}
\][/tex]
[tex]\[
y = \frac{11 \pm 5}{6}
\][/tex]
4. Calculate the roots:
[tex]\[
y_1 = \frac{11 + 5}{6} = \frac{16}{6} = \frac{8}{3}
\][/tex]
[tex]\[
y_2 = \frac{11 - 5}{6} = \frac{6}{6} = 1
\][/tex]
Hence, the solutions to the equation [tex]\(\frac{11y - 2}{3} = y^2 + 2\)[/tex] are:
[tex]\[
y = 1 \quad \text{and} \quad y = \frac{8}{3}
\][/tex]
