To solve the given equation [tex]\( 10 - 9x^2 + 4x = -6x^2 \)[/tex], we will follow these steps:
1. Rewrite the equation in standard quadratic form:
Start by moving all terms to one side of the equation so that we have the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[
10 - 9x^2 + 4x + 6x^2 = 0
\][/tex]
Combine like terms:
[tex]\[
10 - 3x^2 + 4x = 0
\][/tex]
This gives us the standard quadratic form:
[tex]\[
-3x^2 + 4x + 10 = 0
\][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
For the equation [tex]\(-3x^2 + 4x + 10 = 0\)[/tex], we have:
[tex]\[
a = -3, \quad b = 4, \quad c = 10
\][/tex]
3. Determining the discriminant:
The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by [tex]\( \Delta = b^2 - 4ac \)[/tex].
Let's calculate the discriminant:
[tex]\[
\Delta = 4^2 - 4(-3)(10)
\][/tex]
[tex]\[
\Delta = 16 + 120
\][/tex]
[tex]\[
\Delta = 136
\][/tex]
Since the discriminant is positive, we have two distinct real roots.
4. Finding the roots of the quadratic equation:
The roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-4 \pm \sqrt{136}}{2(-3)}
\][/tex]
[tex]\[
x = \frac{-4 \pm \sqrt{136}}{-6}
\][/tex]
Simplify the expression under the square root:
[tex]\[
\sqrt{136} = \sqrt{4 \cdot 34} = 2\sqrt{34}
\][/tex]
So the roots become:
[tex]\[
x = \frac{-4 \pm 2\sqrt{34}}{-6}
\][/tex]
Simplify further by factoring out common terms in the numerator and denominator:
[tex]\[
x = \frac{2(-2 \pm \sqrt{34})}{-6}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{34}}{-3}
\][/tex]
This gives us the solutions:
[tex]\[
x = \frac{2 - \sqrt{34}}{3} \quad \text{and} \quad x = \frac{2 + \sqrt{34}}{3}
\][/tex]
Thus, the correct choice is (D):
[tex]\[
x = \frac{-2 \pm \sqrt{34}}{-3}
\][/tex]
