To solve the quadratic equation \(7x^2 + 3x - 2 = 0\) using the quadratic formula, follow these steps:
1. Identify the coefficients: For the quadratic equation \(ax^2 + bx + c = 0\), the coefficients are:
- \(a = 7\)
- \(b = 3\)
- \(c = -2\)
2. Write down the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant (\(\Delta\)):
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the known values:
[tex]\[
\Delta = 3^2 - 4 \cdot 7 \cdot (-2) = 9 + 56 = 65
\][/tex]
4. Calculate the two solutions using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substituting \(a = 7\), \(b = 3\), and \(\Delta = 65\):
For the first solution (\(x_1\)):
[tex]\[
x_1 = \frac{-3 + \sqrt{65}}{2 \cdot 7}
\][/tex]
Simplifying further:
[tex]\[
x_1 = \frac{-3 + \sqrt{65}}{14} \approx 0.3615898391641821
\][/tex]
For the second solution (\(x_2\)):
[tex]\[
x_2 = \frac{-3 - \sqrt{65}}{2 \cdot 7}
\][/tex]
Simplifying further:
[tex]\[
x_2 = \frac{-3 - \sqrt{65}}{14} \approx -0.7901612677356107
\][/tex]
5. Write the final solutions:
[tex]\[
x_1 \approx 0.3615898391641821
\][/tex]
[tex]\[
x_2 \approx -0.7901612677356107
\][/tex]
Therefore, the solutions to the quadratic equation \(7x^2 + 3x - 2 = 0\) are:
[tex]\[
x \approx 0.3615898391641821 \quad \text{and} \quad x \approx -0.7901612677356107
\][/tex]
