Question: Find the solutions to the following equation by using the quadratic formula.
3x² − 9x + 16 = 178
a.) x = -18 and x = 27
b.) x = -9 and x = 6
c.) x = -6 and x = 9
d.) x = -27 and x = 18
1. **State the problem:** Solve the quadratic equation $$3x^2 - 9x + 16 = 178$$ using the quadratic formula.
2. **Rewrite the equation in standard form:** Move all terms to one side:
$$3x^2 - 9x + 16 - 178 = 0$$
$$3x^2 - 9x - 162 = 0$$
3. **Identify coefficients:**
$$a = 3, \quad b = -9, \quad c = -162$$
4. **Recall the quadratic formula:**
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
5. **Calculate the discriminant:**
$$b^2 - 4ac = (-9)^2 - 4 \times 3 \times (-162) = 81 + 1944 = 2025$$
6. **Find the square root of the discriminant:**
$$\sqrt{2025} = 45$$
7. **Apply the quadratic formula:**
$$x = \frac{-(-9) \pm 45}{2 \times 3} = \frac{9 \pm 45}{6}$$
8. **Calculate the two solutions:**
- For the plus sign:
$$x = \frac{9 + 45}{6} = \frac{54}{6} = 9$$
- For the minus sign:
$$x = \frac{9 - 45}{6} = \frac{-36}{6} = -6$$
9. **Final answer:**
$$x = 9 \quad \text{or} \quad x = -6$$
10. **Match with given options:** Option c.) $x = -6$ and $x = 9$ is correct.