To solve the equation 2s² + 12s – 132 = 0, we can start by simplifying the equation.
1. **Divide the entire equation by 2** to make the coefficients smaller:
s² + 6s - 66 = 0
2. **Now, we can use the quadratic formula** to find the values of s. The quadratic formula is:
s = \frac{-b \pm \sqrt{b² - 4ac}}{2a}
In our case, the coefficients are:
- a = 1
- b = 6
- c = -66
3. **Substitute the values of a, b, and c into the quadratic formula**:
s = \frac{-6 \pm \sqrt{6² - 4 \cdot 1 \cdot (-66)}}{2 \cdot 1}
4. **Calculate the discriminant**:
6² - 4 \cdot 1 \cdot (-66) = 36 + 264 = 300
5. **Now calculate s**:
s = \frac{-6 \pm \sqrt{300}}{2}
6. **Simplifying further, we find the square root of 300**:
\sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}
7. **Plugging this back into the formula gives us**:
s = \frac{-6 \pm 10\sqrt{3}}{2} = -3 \pm 5\sqrt{3}
8. **This results in two possible solutions for s**:
s = -3 + 5\sqrt{3} \quad \text{and} \quad s = -3 - 5\sqrt{3}
9. **Conclusion**: Hence, the equation 2s² + 12s – 132 = 0 has two solutions: s = -3 + 5√3 and s = -3 – 5√3.