To solve the quadratic equation [tex]\(3x^2 + 5x - 2 = 0\)[/tex], we can use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Given the quadratic equation [tex]\(3x^2 + 5x - 2 = 0\)[/tex], we identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] as follows:
[tex]\[
a = 3, \quad b = 5, \quad c = -2
\][/tex]
### Step-by-Step Solution:
1. Calculate the Discriminant: The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(b^2 - 4ac\)[/tex]. Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 5^2 - 4 \cdot 3 \cdot (-2)
\][/tex]
[tex]\[
\text{Discriminant} = 25 + 24 = 49
\][/tex]
2. Apply the Quadratic Formula: The quadratic formula provides the solutions for [tex]\(x\)[/tex] as:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substituting the values of [tex]\(b\)[/tex], the discriminant, and [tex]\(a\)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{49}}{2 \cdot 3}
\][/tex]
[tex]\[
x = \frac{-5 \pm 7}{6}
\][/tex]
3. Calculate the Two Solutions:
- First Solution:
[tex]\[
x_1 = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3}
\][/tex]
- Second Solution:
[tex]\[
x_2 = \frac{-5 - 7}{6} = \frac{-12}{6} = -2
\][/tex]
### Conclusion:
The solutions to the quadratic equation [tex]\(3x^2 + 5x - 2 = 0\)[/tex] are:
[tex]\[
x = \frac{1}{3} \quad \text{or} \quad x = -2
\][/tex]
These steps show the detailed process of solving a quadratic equation using the quadratic formula.
