To solve the quadratic equation [tex]\(-3x^2 + 8x - 3 = 8\)[/tex], we can follow a step-by-step approach.
1. Rearrange the equation:
First, we move all terms to one side to form a standard quadratic equation:
[tex]\[
-3x^2 + 8x - 3 - 8 = 0
\][/tex]
Simplifying, we get:
[tex]\[
-3x^2 + 8x - 11 = 0
\][/tex]
2. Identify coefficients:
From the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
a = -3,\quad b = 8,\quad c = -11
\][/tex]
3. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as follows:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 8^2 - 4(-3)(-11)
\][/tex]
Simplifying further:
[tex]\[
\Delta = 64 - 132
\][/tex]
[tex]\[
\Delta = -68
\][/tex]
Since the discriminant is negative, the solutions will be complex (imaginary).
5. Find the square root of the discriminant:
The square root of the discriminant is:
[tex]\[
\sqrt{-68} = \sqrt{68}i = \sqrt{4 \cdot 17}i = 2\sqrt{17}i
\][/tex]
6. Substitute into the quadratic formula:
Now we substitute the values back into the quadratic formula:
[tex]\[
x = \frac{-8 \pm 2\sqrt{17}i}{2(-3)}
\][/tex]
Simplify the expression:
[tex]\[
x = \frac{-8 \pm 2\sqrt{17}i}{-6}
\][/tex]
Separate the real and imaginary parts:
[tex]\[
x = \frac{-8}{-6} \pm \frac{2\sqrt{17}i}{-6}
\][/tex]
Simplify each part:
[tex]\[
x = \frac{4}{3} \mp \frac{\sqrt{17}i}{3}
\][/tex]
7. Round solutions to three decimal places:
As [tex]\(\sqrt{17} \approx 4.123\)[/tex], we get:
[tex]\[
x = \frac{4}{3} \mp \frac{4.123i}{3}
\][/tex]
Simplify the final solution:
[tex]\[
x = 1.333 \mp 1.374i
\][/tex]
Therefore, the solutions are:
[tex]\[
x = 1.333 + 1.374i \quad \text{and} \quad x = 1.333 - 1.374i
\][/tex]
These are the rounded solutions to three decimal places.
