To find the roots of the quadratic equation [tex]\(-10x^2 + 12x - 9 = 0\)[/tex], we can use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are:
[tex]\(a = -10\)[/tex]
[tex]\(b = 12\)[/tex]
[tex]\(c = -9\)[/tex]
First, let's calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the given values into the discriminant formula:
[tex]\[
\Delta = 12^2 - 4(-10)(-9)
\][/tex]
[tex]\[
\Delta = 144 - 360
\][/tex]
[tex]\[
\Delta = -216
\][/tex]
The discriminant is negative ([tex]\(\Delta = -216\)[/tex]), which indicates that the roots are complex numbers.
Now, we find the roots using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(\Delta = -216\)[/tex], [tex]\(a = -10\)[/tex], and [tex]\(b = 12\)[/tex] into the formula:
[tex]\[
x = \frac{-12 \pm \sqrt{-216}}{2 \cdot (-10)}
\][/tex]
Since the discriminant is negative, we can write [tex]\(\sqrt{-216}\)[/tex] as [tex]\(i\sqrt{216}\)[/tex]:
[tex]\[
x = \frac{-12 \pm i\sqrt{216}}{-20}
\][/tex]
Simplify [tex]\(\sqrt{216}\)[/tex]:
[tex]\[
\sqrt{216} = \sqrt{36 \cdot 6} = 6\sqrt{6}
\][/tex]
Now, substitute [tex]\(\sqrt{216}\)[/tex] into the equation:
[tex]\[
x = \frac{-12 \pm 6i\sqrt{6}}{-20}
\][/tex]
We can simplify the fraction:
[tex]\[
x = \frac{12}{20} \pm \frac{6i\sqrt{6}}{20}
\][/tex]
[tex]\[
x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}
\][/tex]
Thus, the roots of the quadratic equation [tex]\(-10x^2 + 12x - 9 = 0\)[/tex] are:
[tex]\[
x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(x = \frac{3}{5} \pm \frac{3i\sqrt{6}}{10}\)[/tex]
