To solve the quadratic equation [tex]\(2x^2 + 10x - 6 = 0\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
The quadratic equation is in the form [tex]\(ax^2 + bx + c = 0\)[/tex].
[tex]\[
a = 2, \quad b = 10, \quad c = -6
\][/tex]
2. Calculate the discriminant ([tex]\(\Delta\)[/tex]):
The discriminant is given by the formula [tex]\(\Delta = b^2 - 4ac\)[/tex].
[tex]\[
\Delta = 10^2 - 4 \times 2 \times (-6)
\][/tex]
[tex]\[
\Delta = 100 + 48
\][/tex]
[tex]\[
\Delta = 148
\][/tex]
3. Apply the quadratic formula:
The roots are given by the formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values we have:
[tex]\[
x = \frac{-10 \pm \sqrt{148}}{2 \times 2}
\][/tex]
[tex]\[
x = \frac{-10 \pm \sqrt{148}}{4}
\][/tex]
4. Calculate the two roots:
[tex]\[
x_1 = \frac{-10 + \sqrt{148}}{4}
\][/tex]
[tex]\[
x_2 = \frac{-10 - \sqrt{148}}{4}
\][/tex]
Using a calculator to find [tex]\(\sqrt{148} \approx 12.17\)[/tex], we then find:
[tex]\[
x_1 = \frac{-10 + 12.17}{4} \approx \frac{2.17}{4} \approx 0.54
\][/tex]
[tex]\[
x_2 = \frac{-10 - 12.17}{4} \approx \frac{-22.17}{4} \approx -5.54
\][/tex]
5. Round the results to the nearest hundredths place:
[tex]\[
x_1 \approx 0.54
\][/tex]
[tex]\[
x_2 \approx -5.54
\][/tex]
6. Conclusion:
The solutions to the equation [tex]\(2x^2 + 10x - 6 = 0\)[/tex] are:
[tex]\[
x = 0.54 \quad \text{and} \quad x = -5.54
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
\boxed{-5.54 \text{ and } 0.54}
\][/tex]
