Question: Solve for \( y \). \[ (y + 7)^2 = 2y^2 + 12y + 25 \] If there is more than one solution,…

Solve for \( y \).

\[ (y + 7)^2 = 2y^2 + 12y + 25 \]

If there is more than one solution, separate them with commas.

\( y = \boxed{\phantom{0}} \)

Solution

To solve the equation \( (y+7)^2 = 2y^2 + 12y + 25 \), start by expanding both sides: Expand the left side: \[ (y+7)^2 = y^2 + 14y + 49 \] Set the equation equal to the right side: \[ y^2 + 14y + 49 = 2y^2 + 12y + 25 \] Rearrange terms to one side to form a quadratic equation: \[ y^2 + 14y + 49 - 2y^2 - 12y - 25 = 0 \] Combine like terms: \[ -y^2 + 2y + 24 = 0 \] Multiply the entire equation by \(-1\) to make the coefficient of \(y^2\) positive: \[ y^2 - 2y - 24 = 0 \] Factor the quadratic equation: \[ (y - 6)(y + 4) = 0 \] Set each factor to zero and solve for \(y\): \[ \begin{cases} y - 6 = 0 & \\ y + 4 = 0 & \end{cases} \] Solve each equation: \[ \begin{cases} y = 6 & \\ y = -4 & \end{cases} \] Thus, the solutions are \(y = 6\) and \(y = -4\).