Solve for $$ y $$. $$ 2 y^{2}+11 y+5=(y+5)^{2} $$ If there is more than one solution, separate them with commas.

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
2 y^{2} + 11 y + 5 = (y + 5)^{2}
$$

1. **Expand the Right Side:**

$$
(y + 5)^{2} = y^{2} + 10y + 25
$$

2. **Rearrange the Equation:**

$$
2 y^{2} + 11 y + 5 = y^{2} + 10 y + 25
$$

Subtract $$ y^{2} + 10 y + 25 $$ from both sides:

$$
2 y^{2} + 11 y + 5 - y^{2} - 10 y - 25 = 0
$$

Simplify:

$$
y^{2} + y - 20 = 0
$$

3. **Solve the Quadratic Equation:**

Use the quadratic formula $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$, where $$ a = 1 $$, $$ b = 1 $$, and $$ c = -20 $$:

$$
y = \frac{-1 \pm \sqrt{1 + 80}}{2} = \frac{-1 \pm \sqrt{81}}{2} = \frac{-1 \pm 9}{2}
$$

This gives two solutions:

$$
y = \frac{8}{2} = 4 \quad \text{and} \quad y = \frac{-10}{2} = -5
$$

4. **Verify the Solutions:**

- For $$ y = 4 $$:

$$
2(4)^2 + 11(4) + 5 = 32 + 44 + 5 = 81
$$
$$
(4 + 5)^2 = 81
$$

- For $$ y = -5 $$:

$$
2(-5)^2 + 11(-5) + 5 = 50 - 55 + 5 = 0
$$
$$
(-5 + 5)^2 = 0
$$

Both solutions satisfy the original equation.

**Final Answer:**

4, -5
The solutions are $$ y = 4 $$ and $$ y = -5 $$.

Solve for \( y \). \[ 2 y^{2}+11 y+5=(y+5)^{2} \] If there is more than one solution, separate them with commas.