Answer :
Fatengdnhhtest(fhmjfgcjfgccjyfcyfciyfcfjyvdcv kuggcu*vougwbvouhb To solve the quadratic equation \(9x^2 - 7x + 2 = 0\) using the quadratic formula, we use the standard form \(ax^2 + bx + c = 0\) and the formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the given equation \(9x^2 - 7x + 2 = 0\):
- \(a = 9\)
- \(b = -7\)
- \(c = 2\)
Let's plug these values into the quadratic formula:
1. Calculate the discriminant \(\Delta\):
\[
\Delta = b^2 - 4ac = (-7)^2 - 4 \cdot 9 \cdot 2 = 49 - 72 = -23
\]
Since the discriminant is negative (\(\Delta = -23\)), the equation has no real roots, but it has two complex roots. We can proceed to find these complex roots:
2. Calculate the roots using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-7) \pm \sqrt{-23}}{2 \cdot 9} = \frac{7 \pm \sqrt{-23}}{18}
\]
3. Simplify the roots. Since \(\sqrt{-23} = i\sqrt{23}\):
\[
x = \frac{7 \pm i\sqrt{23}}{18}
\]
So, the solutions to the quadratic equation \(9x^2 - 7x + 2 = 0\) are:
\[
x = \frac{7 + i\sqrt{23}}{18} \quad \text{and} \quad x = \frac{7 - i\sqrt{23}}{18}
\]
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the given equation \(9x^2 - 7x + 2 = 0\):
- \(a = 9\)
- \(b = -7\)
- \(c = 2\)
Let's plug these values into the quadratic formula:
1. Calculate the discriminant \(\Delta\):
\[
\Delta = b^2 - 4ac = (-7)^2 - 4 \cdot 9 \cdot 2 = 49 - 72 = -23
\]
Since the discriminant is negative (\(\Delta = -23\)), the equation has no real roots, but it has two complex roots. We can proceed to find these complex roots:
2. Calculate the roots using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-(-7) \pm \sqrt{-23}}{2 \cdot 9} = \frac{7 \pm \sqrt{-23}}{18}
\]
3. Simplify the roots. Since \(\sqrt{-23} = i\sqrt{23}\):
\[
x = \frac{7 \pm i\sqrt{23}}{18}
\]
So, the solutions to the quadratic equation \(9x^2 - 7x + 2 = 0\) are:
\[
x = \frac{7 + i\sqrt{23}}{18} \quad \text{and} \quad x = \frac{7 - i\sqrt{23}}{18}
\]