To solve the quadratic equation [tex]\( 8x^2 - 7x + 1 = 0 \)[/tex] using the quadratic formula, we follow these steps:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 8 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = 1 \)[/tex].
1. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = (-7)^2 - 4 \cdot 8 \cdot 1 = 49 - 32 = 17
\][/tex]
2. Calculate the two solutions using the quadratic formula:
[tex]\[
x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a}, \quad x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant:
[tex]\[
x_1 = \frac{-(-7) + \sqrt{17}}{2 \cdot 8} = \frac{7 + \sqrt{17}}{16}
\][/tex]
[tex]\[
x_2 = \frac{-(-7) - \sqrt{17}}{2 \cdot 8} = \frac{7 - \sqrt{17}}{16}
\][/tex]
3. Calculate the exact values of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x_1 \approx \frac{7 + 4.1231}{16} \approx \frac{11.1231}{16} \approx 0.6952
\][/tex]
[tex]\[
x_2 \approx \frac{7 - 4.1231}{16} \approx \frac{2.8769}{16} \approx 0.1798
\][/tex]
4. Round the solutions to the nearest hundredth:
[tex]\[
x_1 \approx 0.70
\][/tex]
[tex]\[
x_2 \approx 0.18
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 8x^2 - 7x + 1 = 0 \)[/tex], rounded to the nearest hundredth, are:
[tex]\[
x \approx 0.70, 0.18
\][/tex]
