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.

Solution

To solve the equation \((y+7)^2 = 2y^2 + 12y + 25\) for \(y\), we will first expand and simplify both sides. Expand the left side: \[ (y+7)^2 = y^2 + 14y + 49 \] Now, the equation becomes: \[ y^2 + 14y + 49 = 2y^2 + 12y + 25 \] Subtract \(y^2\), \(14y\), and \(49\) from both sides to set the equation to zero: \[ 0 = 2y^2 + 12y + 25 - y^2 - 14y - 49 \] Simplify the equation: \[ 0 = y^2 - 2y - 24 \] Now, factor the quadratic equation: \[ 0 = (y - 6)(y + 4) \] Set each factor equal to zero and solve for \(y\): \[ \begin{cases} y - 6 = 0 & \\ y + 4 = 0 & \end{cases} \] Solve the equations: \[ \begin{cases} y = 6 & \\ y = -4 & \end{cases} \] Thus, the solutions are \(y = 6\) and \(y = -4\).