To solve the quadratic equation [tex]\( 0 = -3x^2 - 2x + 6 \)[/tex] using the quadratic formula, we need to identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the equation in the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Here, we have:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -2 \)[/tex]
- [tex]\( c = 6 \)[/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the given values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
1. Substitute [tex]\( b = -2 \)[/tex]:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This simplifies to:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 72}}{-6} \][/tex]
Then,
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
So the correct substitution of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] in the quadratic formula is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This matches:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This shows the correct substitution of the values [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the equation [tex]\( 0 = -3x^2 - 2x + 6 \)[/tex] into the quadratic formula.
