Let's solve the quadratic equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex] using the quadratic formula:
The quadratic formula is given by:
[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] in the quadratic equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex]:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -8\)[/tex]
- [tex]\(c = 7\)[/tex]
Next, we will calculate the discriminant, which is given by:
[tex]\[
\text{discriminant} = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we get:
[tex]\[
\text{discriminant} = (-8)^2 - 4 \cdot 2 \cdot 7 = 64 - 56 = 8
\][/tex]
Now, we will calculate the two possible solutions using the quadratic formula:
[tex]\[
x_{1,2} = \frac{-b \pm \sqrt{\text{discriminant}}}{2a}
\][/tex]
Substituting the values:
[tex]\[
x_{1,2} = \frac{-(-8) \pm \sqrt{8}}{2 \cdot 2} = \frac{8 \pm \sqrt{8}}{4}
\][/tex]
We simplify the expression under the square root and further solve for [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x_1 = \frac{8 + \sqrt{8}}{4}
\][/tex]
[tex]\[
x_2 = \frac{8 - \sqrt{8}}{4}
\][/tex]
Finally, to get the values rounded to the nearest hundredths place, we proceed with the computation:
After calculating:
[tex]\[
x_1 \approx 2.71
\][/tex]
[tex]\[
x_2 \approx 1.29
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(2x^2 - 8x + 7 = 0\)[/tex] rounded to the nearest hundredths place are:
[tex]\[
x = 1.29 \quad \text{and} \quad x = 2.71
\][/tex]
So, the correct answer is:
[tex]\[
x = 1.29 \quad \text{and} \quad x = 2.71
\][/tex]
