To solve the quadratic equation \(2x^2 - 3x - 8 = 0\) using the quadratic formula, we apply the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a\), \(b\), and \(c\) are the coefficients from the equation \(ax^2 + bx + c = 0\).
For the equation \(2x^2 - 3x - 8 = 0\):
- \(a = 2\)
- \(b = -3\)
- \(c = -8\)
Now we can substitute these values into the formula:
- Calculate \(b^2 - 4ac\):
\[ b^2 = (-3)^2 = 9 \] \[ 4ac = 4 \cdot 2 \cdot (-8) = -64 \] \[ b^2 - 4ac = 9 - (-64) = 9 + 64 = 73 \]
- Now substitute back into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{73}}{2 \cdot 2} \] \[ x = \frac{3 \pm \sqrt{73}}{4} \]
- Calculate the two possible values for \(x\):
- For \(x_1\):
\[ x_1 = \frac{3 + \sqrt{73}}{4} \]
- For \(x_2\):
\[ x_2 = \frac{3 - \sqrt{73}}{4} \]
- Now, we will approximate the values of \(x_1\) and \(x_2\):
Calculating \(\sqrt{73} \approx 8.544\):
- \(x_1\):
\[ x_1 \approx \frac{3 + 8.544}{4} = \frac{11.544}{4} \approx 2.886 \]
- \(x_2\):
\[ x_2 \approx \frac{3 - 8.544}{4} = \frac{-5.544}{4} \approx -1.386 \]
Therefore, the solutions to the equation \(2x^2 - 3x - 8 = 0\) rounded to the nearest hundredth are:
\[ x \approx 2.89, \quad x \approx -1.39 \]
Thus, the final answer is:
\[ x = 2.89 \quad \text{and} \quad x = -1.39 \]