To solve the quadratic equation [tex]\(6z^2 - z - 2 = 0\)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients:
In the quadratic equation [tex]\(az^2 + bz + c = 0\)[/tex], we can identify the coefficients as:
[tex]\[
a = 6, \quad b = -1, \quad c = -2
\][/tex]
2. Write down the quadratic formula:
The quadratic formula for solving [tex]\(az^2 + bz + c = 0\)[/tex] is given by:
[tex]\[
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is [tex]\(b^2 - 4ac\)[/tex]. Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-1)^2 - 4(6)(-2) = 1 + 48 = 49
\][/tex]
4. Find the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{49} = 7
\][/tex]
5. Substitute the values into the quadratic formula:
[tex]\[
z = \frac{-(-1) \pm 7}{2(6)}
\][/tex]
Simplifying further:
[tex]\[
z = \frac{1 \pm 7}{12}
\][/tex]
6. Solve for the two possible values of [tex]\(z\)[/tex]:
- For [tex]\(z_1\)[/tex]:
[tex]\[
z_1 = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3}
\][/tex]
- For [tex]\(z_2\)[/tex]:
[tex]\[
z_2 = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2}
\][/tex]
Thus, the solution set for the quadratic equation [tex]\(6z^2 - z - 2 = 0\)[/tex] is:
[tex]\[
\left\{ \frac{2}{3}, -\frac{1}{2} \right\}
\][/tex]
