To solve the quadratic equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex] using the quadratic formula, we need to use the given formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
First, let's identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the equation:
[tex]\[a = 2\][/tex]
[tex]\[b = -8\][/tex]
[tex]\[c = 7\][/tex]
Next, we calculate the discriminant, which is given by [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\text{Discriminant} = (-8)^2 - 4 \cdot 2 \cdot 7
\][/tex]
[tex]\[
= 64 - 56
\][/tex]
[tex]\[
= 8
\][/tex]
Now, we apply the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substituting in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant:
[tex]\[
x = \frac{-(-8) \pm \sqrt{8}}{2 \cdot 2}
\][/tex]
Simplifying the numerator and denominator:
[tex]\[
x = \frac{8 \pm \sqrt{8}}{4}
\][/tex]
The square root of 8 can be simplified further:
[tex]\[
\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}
\][/tex]
Substituting this back into the equation:
[tex]\[
x = \frac{8 \pm 2\sqrt{2}}{4}
\][/tex]
This can be simplified by dividing each part of the numerator by 4:
[tex]\[
x = \frac{8}{4} \pm \frac{2\sqrt{2}}{4}
\][/tex]
[tex]\[
x = 2 \pm \frac{\sqrt{2}}{2}
\][/tex]
We need the approximate numerical values rounded to the nearest hundredths.
Calculating the two possible values:
[tex]\[
x_1 = 2 + \frac{\sqrt{2}}{2} \approx 2.71
\][/tex]
[tex]\[
x_2 = 2 - \frac{\sqrt{2}}{2} \approx 1.29
\][/tex]
Thus, the solutions to the equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex] are:
[tex]\[ x = 2.71 \quad \text{and} \quad x = 1.29 \][/tex]
Therefore, the correct answer is:
[tex]\[ x = 1.29 \quad \text{and} \quad x = 2.71 \][/tex]
