Y²=4y+96

To solve the equation \( y^2 = 4y + 96 \), we can first rearrange it into standard quadratic form. 1. **Rearrange the equation:** \[ y^2 - 4y - 96 = 0 \] 2. **Now we can factor the quadratic, use the quadratic formula, or complete the square. Let's use the quadratic formula**, where \( a = 1 \), \( b = -4 \), and \( c = -96 \). The quadratic formula is given by: \[ y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Plugging in the values: \[ y = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4(1)(-96)}}}}{2(1)} \] \[ y = \frac{{4 \pm \sqrt{{16 + 384}}}}{2} \] \[ y = \frac{{4 \pm \sqrt{400}}}{2} \] \[ y = \frac{{4 \pm 20}}{2} \] 3. **Calculate the possible values for \( y \):** - For the positive case: \[ y = \frac{24}{2} = 12 \] - For the negative case: \[ y = \frac{-16}{2} = -8 \] 4. **Conclusion:** The solutions to the equation \( y^2 = 4y + 96 \) are: \[ y = 12 \quad \text{and} \quad y = -8 \]