To solve the quadratic equation [tex]\( 12x^2 - 22x - 20 = 0 \)[/tex] using the quadratic formula, we follow these steps:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
First, identify the coefficients in the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[
a = 12, \quad b = -22, \quad c = -20
\][/tex]
Next, calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
[tex]\[
\text{Discriminant} = (-22)^2 - 4 \cdot 12 \cdot (-20)
\][/tex]
[tex]\[
\text{Discriminant} = 484 + 960
\][/tex]
[tex]\[
\text{Discriminant} = 1444
\][/tex]
Since the discriminant is positive ([tex]\(1444\)[/tex]), there are two distinct real solutions. We then use the quadratic formula to find these solutions:
[tex]\[
x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a}, \quad x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a}
\][/tex]
[tex]\[
x_1 = \frac{-(-22) + \sqrt{1444}}{2 \cdot 12}, \quad x_2 = \frac{-(-22) - \sqrt{1444}}{2 \cdot 12}
\][/tex]
[tex]\[
x_1 = \frac{22 + 38}{24}, \quad x_2 = \frac{22 - 38}{24}
\][/tex]
[tex]\[
x_1 = \frac{60}{24}, \quad x_2 = \frac{-16}{24}
\][/tex]
[tex]\[
x_1 = 2.5, \quad x_2 = -\frac{2}{3}
\][/tex]
Thus, the two solutions are:
[tex]\[
\text{The smaller solution is } x = -\frac{2}{3}
\][/tex]
[tex]\[
\text{The larger solution is } x = 2.5
\][/tex]
Therefore, the answers are:
[tex]\[
\text{The smaller solution is } x = -\frac{2}{3} \quad \text{(in fraction form, if possible)}
\][/tex]
[tex]\[
\text{The larger solution is } x = 2.5
\][/tex]
