To solve the quadratic equation [tex]\(-3 x^2 + 30 x - 90 = 0\)[/tex], we need to determine the roots of the equation. Here are the steps to do so:
1. Identify the coefficients: In the given quadratic equation [tex]\( -3 x^2 + 30 x - 90 = 0 \)[/tex], the coefficients are:
- [tex]\(a = -3\)[/tex]
- [tex]\(b = 30\)[/tex]
- [tex]\(c = -90\)[/tex]
2. Form the quadratic equation in standard form: The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. We already have our equation in this form.
3. Calculate the discriminant ([tex]\(\Delta\)[/tex]): The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 30^2 - 4(-3)(-90) = 900 - 1080 = -180
\][/tex]
4. Find the roots using the quadratic formula: The quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex] is used to find the roots of the equation. Since the discriminant [tex]\(\Delta\)[/tex] is negative, we will have complex roots. The formula is modified for complex roots as:
[tex]\[
x = \frac{-b \pm \sqrt{-\Delta} \cdot i}{2a}
\][/tex]
Substituting the known values:
[tex]\[
x = \frac{-30 \pm \sqrt{180} \cdot i}{2(-3)}
\][/tex]
5. Simplify the expression:
[tex]\[
\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}
\][/tex]
Therefore:
[tex]\[
x = \frac{-30 \pm 6\sqrt{5} \cdot i}{-6} = \frac{30 \mp 6\sqrt{5} \cdot i}{6} = 5 \mp \sqrt{5} \cdot i
\][/tex]
Thus, the roots of the quadratic equation [tex]\(-3 x^2 + 30 x - 90 = 0\)[/tex] are:
[tex]\[
x = 5 - \sqrt{5} \cdot i \quad \text{and} \quad x = 5 + \sqrt{5} \cdot i
\][/tex]
Therefore, the correct answer is:
\[
x = 5 \pm i \sqrt{5}
\
