To solve the equation ( y^2 = 4y + 96 ), we can first rearrange it to the standard form of a quadratic equation. This involves moving all terms to one side of the equation:
[
y^2 - 4y - 96 = 0
]
Now, we can solve this quadratic equation using the quadratic formula:
[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
In our case, ( a = 1 ), ( b = -4 ), and ( c = -96 ).
- First, we calculate the discriminant ( b^2 - 4ac ):
[
b^2 = (-4)^2 = 16
]
[
4ac = 4 \cdot 1 \cdot (-96) = -384
]
[
b^2 - 4ac = 16 - (-384) = 16 + 384 = 400
]
- Now we substitute the values into the quadratic formula:
[
y = \frac{-(-4) \pm \sqrt{400}}{2 \cdot 1}
]
[
y = \frac{4 \pm 20}{2}
]
- This gives us two possible solutions:
[
y_1 = \frac{4 + 20}{2} = \frac{24}{2} = 12
]
[
y_2 = \frac{4 - 20}{2} = \frac{-16}{2} = -8
]
Thus, the solutions to the quadratic equation ( y^2 = 4y + 96 ) are:
[
y = 12 \quad \text{and} \quad y = -8
]
