To solve the quadratic equation [tex]\(-10x^2 + 12x - 9 = 0\)[/tex], we will use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = -10\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(c = -9\)[/tex].
First, we calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = 12^2 - 4(-10)(-9)
\][/tex]
[tex]\[
\text{Discriminant} = 144 - 360
\][/tex]
[tex]\[
\text{Discriminant} = -216
\][/tex]
Since the discriminant is negative ([tex]\(-216\)[/tex]), the roots will be complex numbers.
Next, we calculate the roots using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute [tex]\(b\)[/tex], the discriminant, and [tex]\(a\)[/tex]:
[tex]\[
x = \frac{-12 \pm \sqrt{-216}}{2(-10)}
\][/tex]
First, simplify [tex]\(\sqrt{-216}\)[/tex]:
[tex]\[
\sqrt{-216} = \sqrt{-1 \cdot 216} = i\sqrt{216} = i\sqrt{36 \cdot 6} = i \cdot 6 \cdot \sqrt{6} = 6i\sqrt{6}
\][/tex]
Now we need to put it back into the formula:
[tex]\[
x = \frac{-12 \pm 6i\sqrt{6}}{-20}
\][/tex]
Simplify the fractions by dividing the terms by the common factor:
[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 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:
A. [tex]\(\boxed{x = \frac{3}{5} \pm \frac{3 i \sqrt{6}}{10}}\)[/tex]
