To solve the quadratic equation [tex]\(-6x^2 - 4x - 2 = -9\)[/tex], follow these steps:
1. Rewrite the equation in standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex]:
Start by moving [tex]\(-9\)[/tex] to the left side of the equation to set the equation to zero:
[tex]\[
-6x^2 - 4x - 2 + 9 = 0
\][/tex]
Simplify this to:
[tex]\[
-6x^2 - 4x + 7 = 0
\][/tex]
2. Identify the coefficients:
In the equation [tex]\(-6x^2 - 4x + 7 = 0\)[/tex], we have:
[tex]\[
a = -6, \quad b = -4, \quad c = 7
\][/tex]
3. Compute the discriminant using the formula [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = (-4)^2 - 4(-6)(7) = 16 + 168 = 184
\][/tex]
The discriminant is [tex]\(184\)[/tex].
4. Compute the solutions using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{\Delta}}{2a}\)[/tex]:
[tex]\[
x = \frac{-(-4) \pm \sqrt{184}}{2(-6)} = \frac{4 \pm \sqrt{184}}{-12}
\][/tex]
5. Find the two solutions:
- The first solution [tex]\(x_1\)[/tex] is:
[tex]\[
x_1 = \frac{4 + \sqrt{184}}{-12} \approx \frac{4 + 13.565}{-12} = \frac{17.565}{-12} \approx -1.464
\][/tex]
- The second solution [tex]\(x_2\)[/tex] is:
[tex]\[
x_2 = \frac{4 - \sqrt{184}}{-12} \approx \frac{4 - 13.565}{-12} = \frac{-9.565}{-12} \approx 0.797
\][/tex]
Therefore, the solutions to the equation [tex]\(-6x^2 - 4x + 7 = 0\)[/tex] are approximately [tex]\(x_1 \approx -1.464\)[/tex] and [tex]\(x_2 \approx 0.797\)[/tex], rounded to three decimal places.
