To solve the equation [tex]\(2 y^2 - 3 y + 5 = (y - 3)^2\)[/tex], follow these steps:
1. Expand the right-hand side:
[tex]\[
(y - 3)^2 = y^2 - 6y + 9
\][/tex]
2. Substitute the expanded form back into the equation:
[tex]\[
2 y^2 - 3 y + 5 = y^2 - 6y + 9
\][/tex]
3. Move all terms to one side of the equation to set it to zero:
[tex]\[
2 y^2 - 3 y + 5 - y^2 + 6y - 9 = 0
\][/tex]
4. Combine like terms:
[tex]\[
2 y^2 - y^2 - 3 y + 6 y + 5 - 9 = 0
\][/tex]
[tex]\[
y^2 + 3y - 4 = 0
\][/tex]
5. Factorize the quadratic equation:
[tex]\[
y^2 + 3y - 4 = (y + 4)(y - 1) = 0
\][/tex]
6. Set each factor equal to zero and solve for [tex]\(y\)[/tex]:
[tex]\[
y + 4 = 0 \quad \text{or} \quad y - 1 = 0
\][/tex]
Solving these gives:
[tex]\[
y = -4 \quad \text{or} \quad y = 1
\][/tex]
Therefore, the solutions for [tex]\(y\)[/tex] are:
[tex]\[
y = -4, 1
\][/tex]
