Question

Use the quadratic formula to solve 2x2−3x−8=0
2
𝑥
2
−
3
𝑥
−
8
=
0


Round your answer to the nearest hundredth if necessary.

Reference Sheet - Quadratic Formula

(2 points)
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x=
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Answer 1

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:

1. Calculate \(b^2 - 4ac\):

\[
b^2 = (-3)^2 = 9
\]
\[
4ac = 4 \cdot 2 \cdot (-8) = -64
\]
\[
b^2 - 4ac = 9 - (-64) = 9 + 64 = 73
\]

2. Now substitute back into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{73}}{2 \cdot 2}
\]
\[
x = \frac{3 \pm \sqrt{73}}{4}
\]

3. Calculate the two possible values for \(x\):

- For \(x_1\):

\[
x_1 = \frac{3 + \sqrt{73}}{4}
\]

- For \(x_2\):

\[
x_2 = \frac{3 - \sqrt{73}}{4}
\]

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:

\[
x = 2.89 \quad \text{and} \quad x = -1.39
\]