To find the correct substitution of the values \(a = -3\), \(b = -2\), and \(c = 6\) into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we'll follow these steps:
1. Identify the values of \(a\), \(b\), and \(c\):
- \(a = -3\)
- \(b = -2\)
- \(c = 6\)
2. Substitute these values into the quadratic formula.
Remember the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substitute \(b = -2\), \(a = -3\), and \(c = 6\):
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This correctly includes the values from the equation \(0 = -3x^2 - 2x + 6\). Let's break down what each part represents:
- \(-b\) becomes \(-(-2)\)
- \(b^2\) becomes \((-2)^2\)
- \(-4ac\) becomes \(-4(-3)(6)\)
- \(2a\) becomes \(2(-3)\)
3. Evaluate the equation inside the square root and the denominator:
[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]
Thus, the correct substitution of the values \(a = -3\), \(b = -2\), and \(c = 6\) into the quadratic formula is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
