To solve the quadratic equation [tex]\(0 = -3x^2 - 4x + 5\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients: In the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], we have:
- [tex]\(a = -3\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 5\)[/tex]
2. Calculate the discriminant: The discriminant ([tex]\(\Delta\)[/tex]) is given by [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = (-4)^2 - 4 \cdot (-3) \cdot 5 = 16 + 60 = 76
\][/tex]
The discriminant is 76.
3. Apply the quadratic formula: The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
x = \frac{-(-4) \pm \sqrt{76}}{2 \cdot (-3)} = \frac{4 \pm \sqrt{76}}{-6}
\][/tex]
4. Simplify the square root and the fraction: Firstly, we simplify [tex]\(\sqrt{76}\)[/tex]:
[tex]\[
\sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19}
\][/tex]
Now, plug this back into the formula:
[tex]\[
x = \frac{4 \pm 2\sqrt{19}}{-6}
\][/tex]
5. Simplify the expression: We can factor out a 2 from the numerator:
[tex]\[
x = \frac{2(2 \pm \sqrt{19})}{-6} = \frac{2 \pm \sqrt{19}}{-3}
\][/tex]
Since the fraction has a negative denominator, we can rewrite it as:
[tex]\[
x = -\frac{2 \pm \sqrt{19}}{3}
\][/tex]
Hence, the solutions to the quadratic equation [tex]\(0 = -3x^2 - 4x + 5\)[/tex] in simplest radical form are:
[tex]\[
x = -\frac{2 + \sqrt{19}}{3} \quad \text{and} \quad x = -\frac{2 - \sqrt{19}}{3}
\][/tex]
So the correct answer is:
[tex]\[
x = -\frac{2 \pm \sqrt{19}}{3}
\][/tex]
