To solve the quadratic equation [tex]\(3x^2 - 7x + 2 = 0\)[/tex], we need to use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. In our case, the coefficients are:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = 2\)[/tex]
Let's go through the solution step-by-step:
1. Calculate the Discriminant
The discriminant [tex]\(\Delta\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = (-7)^2 - 4 \cdot 3 \cdot 2
\][/tex]
[tex]\[
\Delta = 49 - 24
\][/tex]
[tex]\[
\Delta = 25
\][/tex]
2. Calculate the Two Solutions
Since the discriminant is positive ([tex]\(\Delta = 25\)[/tex]), we have two distinct real roots. The quadratic formula will give us these roots:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(\Delta = 25\)[/tex], and the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x = \frac{-(-7) \pm \sqrt{25}}{2 \cdot 3}
\][/tex]
[tex]\[
x = \frac{7 \pm 5}{6}
\][/tex]
Now, calculate each root separately:
- For the positive square root:
[tex]\[
x_1 = \frac{7 + 5}{6}
\][/tex]
[tex]\[
x_1 = \frac{12}{6}
\][/tex]
[tex]\[
x_1 = 2
\][/tex]
- For the negative square root:
[tex]\[
x_2 = \frac{7 - 5}{6}
\][/tex]
[tex]\[
x_2 = \frac{2}{6}
\][/tex]
[tex]\[
x_2 = \frac{1}{3}
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\(3x^2 - 7x + 2 = 0\)[/tex] are:
[tex]\[
x_1 = 2
\][/tex]
[tex]\[
x_2 = \frac{1}{3}
\][/tex]
Lastly, the discriminant is [tex]\(\Delta = 25\)[/tex].
Thus, the final answers are:
- The discriminant [tex]\(\Delta = 25\)[/tex]
- The roots are [tex]\(x_1 = 2\)[/tex] and [tex]\(x_2 = \frac{1}{3}\)[/tex]
