Let's solve the quadratic equation [tex]\(0 = -3x^2 - 4x + 5\)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, identify the coefficients from the given quadratic equation:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 5 \)[/tex]
### Calculation of Discriminant
The discriminant ([tex]\(\Delta\)[/tex]) is part of the quadratic formula and is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plug in the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-4)^2 - 4(-3)(5)
\][/tex]
Calculate this step-by-step:
1. Compute [tex]\((-4)^2\)[/tex]:
[tex]\[
(-4)^2 = 16
\][/tex]
2. Compute [tex]\(4(-3)(5)\)[/tex]:
[tex]\[
4 \cdot (-3) \cdot 5 = -60
\][/tex]
3. Subtract these values:
[tex]\[
\Delta = 16 - (-60) = 16 + 60 = 76
\][/tex]
So, the discriminant is:
[tex]\[
\Delta = 76
\][/tex]
### Solving by the Quadratic Formula
Now, apply the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(b\)[/tex], [tex]\(a\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
x = \frac{-(-4) \pm \sqrt{76}}{2(-3)}
\][/tex]
Simplify inside the formula:
1. [tex]\(-(-4) = 4\)[/tex]
2. [tex]\(\sqrt{76}\)[/tex]
[tex]\[
x = \frac{4 \pm \sqrt{76}}{-6}
\][/tex]
Next, express [tex]\(\sqrt{76}\)[/tex] in simplest radical form. Since [tex]\(76 = 4 \cdot 19\)[/tex], we write:
[tex]\[
\sqrt{76} = \sqrt{4 \cdot 19} = 2\sqrt{19}
\][/tex]
Substitute [tex]\(\sqrt{76}\)[/tex] with [tex]\(2\sqrt{19}\)[/tex]:
[tex]\[
x = \frac{4 \pm 2\sqrt{19}}{-6}
\][/tex]
Simplify the fraction by dividing the numerator and the denominator by 2:
[tex]\[
x = \frac{4/2 \pm 2\sqrt{19}/2}{-6/2} = \frac{2 \pm \sqrt{19}}{-3}
\][/tex]
Distribute the negative sign in the denominator:
[tex]\[
x = -\frac{2 \pm \sqrt{19}}{3}
\][/tex]
Thus, the solutions 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 solution is:
[tex]\[ x = -\frac{2 \pm \sqrt{19}}{3} \][/tex]
