To solve the quadratic equation [tex]\(2x^2 - 11x = 2\)[/tex] using the quadratic formula, we first need to rewrite the equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[2x^2 - 11x - 2 = 0\][/tex]
Here, the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -11\)[/tex]
- [tex]\(c = -2\)[/tex]
The quadratic formula is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
1. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is found using the formula:
[tex]\[\Delta = b^2 - 4ac\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\begin{align*}
\Delta & = (-11)^2 - 4 \cdot 2 \cdot (-2) \\
& = 121 + 16 \\
& = 137
\end{align*}
\][/tex]
2. Calculate the two solutions using the quadratic formula:
The quadratic formula has two solutions based on the plus and minus signs:
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a} \quad \text{and} \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
\begin{align*}
x_1 & = \frac{-(-11) + \sqrt{137}}{2 \cdot 2} \\
& = \frac{11 + \sqrt{137}}{4} \\
& \approx \frac{11 + 11.7047}{4} \\
& \approx \frac{22.7047}{4} \\
& \approx 5.6762
\end{align*}
\][/tex]
[tex]\[
\begin{align*}
x_2 & = \frac{-(-11) - \sqrt{137}}{2 \cdot 2} \\
& = \frac{11 - \sqrt{137}}{4} \\
& \approx \frac{11 - 11.7047}{4} \\
& \approx \frac{-0.7047}{4} \\
& \approx -0.1762
\end{align*}
\][/tex]
3. Round the solutions to the nearest hundredth:
Rounding [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x_1 \approx 5.68
\][/tex]
[tex]\[
x_2 \approx -0.18
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(2x^2 - 11x - 2 = 0\)[/tex] are approximately:
[tex]\[x_1 \approx 5.68\][/tex]
[tex]\[x_2 \approx -0.18\][/tex]
