To solve the quadratic equation [tex]\(7x^2 - 9x + 1 = 0\)[/tex] using the quadratic formula, we follow a systematic approach. The quadratic formula is:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Here, the coefficients are:
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -9\)[/tex]
- [tex]\(c = 1\)[/tex]
Step-by-step, let's substitute these values into the quadratic formula.
1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-9)^2 - 4 \cdot 7 \cdot 1 = 81 - 28 = 53
\][/tex]
2. Calculate the Two Solutions:
Using the quadratic formula, we find two solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]:
[tex]\[
x = \frac{{-b \pm \sqrt{\Delta}}}{2a}
\][/tex]
Substituting the values:
[tex]\[
x_1 = \frac{{-(-9) - \sqrt{53}}}{2 \cdot 7} = \frac{9 - \sqrt{53}}{14}
\][/tex]
[tex]\[
x_2 = \frac{{-(-9) + \sqrt{53}}}{2 \cdot 7} = \frac{9 + \sqrt{53}}{14}
\][/tex]
3. Round the Solutions to Two Decimal Places:
- [tex]\(x_1\)[/tex] value: [tex]\(\frac{9 - \sqrt{53}}{14} \approx 0.12\)[/tex]
- [tex]\(x_2\)[/tex] value: [tex]\(\frac{9 + \sqrt{53}}{14} \approx 1.16\)[/tex]
Therefore, the solutions to the quadratic equation [tex]\(7x^2 - 9x + 1 = 0\)[/tex] accurate to two decimal places are:
[tex]\[
\text{smaller } x\text{-value} \quad x = 0.12
\][/tex]
[tex]\[
\text{larger } x\text{-value} \quad x = 1.16
\][/tex]
