To determine the roots of the quadratic equation [tex]\(-10x^2 + 12x - 9 = 0\)[/tex], we will proceed step-by-step using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Step 1: Identify the coefficients.
From the equation [tex]\(-10x^2 + 12x - 9 = 0\)[/tex], the coefficients are:
- [tex]\(a = -10\)[/tex]
- [tex]\(b = 12\)[/tex]
- [tex]\(c = -9\)[/tex]
Step 2: Calculate the discriminant.
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = 12^2 - 4(-10)(-9) = 144 - 360 = -216
\][/tex]
Since the discriminant is negative ([tex]\(\Delta = -216\)[/tex]), the roots will be complex.
Step 3: Calculate the real and imaginary parts of the roots.
The quadratic formula gives the roots:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
First, determine the real part:
[tex]\[
\text{Real part} = \frac{-b}{2a} = \frac{-12}{2(-10)} = \frac{-12}{-20} = 0.6
\][/tex]
Next, determine the imaginary part. Because the discriminant is negative, we need to take the square root of the absolute value of [tex]\(\Delta\)[/tex] and divide by [tex]\(2a\)[/tex]:
[tex]\[
\text{Imaginary part} = \frac{\sqrt{|\Delta|}}{2a} = \frac{\sqrt{216}}{2(-10)} = \frac{\sqrt{216}}{-20}
\][/tex]
Simplify [tex]\(\sqrt{216}\)[/tex]:
[tex]\[
\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}
\][/tex]
So the imaginary part is:
[tex]\[
\frac{6\sqrt{6}}{-20} = -\frac{3\sqrt{6}}{10}
\][/tex]
The roots are:
[tex]\[
x = \text{Real part} \pm \text{Imaginary part}i = 0.6 \pm \left( -\frac{3\sqrt{6}}{10} \right)i
\][/tex]
[tex]\[
x = 0.6 \pm \left( \frac{-3\sqrt{6}}{10} \right)i
\][/tex]
Since the question asks to present the answer in a specific format, it is useful to check the given options. Given our roots [tex]\(0.6 \pm \left( \frac{-3\sqrt{6}}{10} \right)i \)[/tex] can be represented as:
[tex]\[
x = \frac{3}{5} \pm \frac{3 i \sqrt{6}}{10}
\][/tex]
So the correct answer is:
D. [tex]\( x = \frac{3}{5} \pm \frac{3 i \sqrt{6}}{10} \)[/tex]
