Question: Solve the equation $1.8x^2 - 5.9x - 3.4 = 1.4$ by using the calculator.
Input your solutions below in any order and round to three decimal places if necessary.
x_1 = or x_2 =
1. **State the problem:** Solve the quadratic equation $$1.8x^2 - 5.9x - 3.4 = 1.4$$ for $x$.
2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:
$$1.8x^2 - 5.9x - 3.4 - 1.4 = 0$$
$$1.8x^2 - 5.9x - 4.8 = 0$$
3. **Identify coefficients:**
- $a = 1.8$
- $b = -5.9$
- $c = -4.8$
4. **Use the quadratic formula:**
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
5. **Calculate the discriminant:**
$$\Delta = b^2 - 4ac = (-5.9)^2 - 4 \times 1.8 \times (-4.8)$$
$$= 34.81 + 34.56 = 69.37$$
6. **Calculate the square root of the discriminant:**
$$\sqrt{69.37} \approx 8.327$$
7. **Calculate the two solutions:**
$$x_1 = \frac{-(-5.9) + 8.327}{2 \times 1.8} = \frac{5.9 + 8.327}{3.6} = \frac{14.227}{3.6} \approx 3.952$$
$$x_2 = \frac{-(-5.9) - 8.327}{2 \times 1.8} = \frac{5.9 - 8.327}{3.6} = \frac{-2.427}{3.6} \approx -0.674$$
8. **Final answer:**
$$x_1 \approx 3.952, \quad x_2 \approx -0.674$$
These are the solutions rounded to three decimal places.