To solve the quadratic equation [tex]\( 3x^2 + 22x + 7 = 0 \)[/tex], we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation, [tex]\(a = 3\)[/tex], [tex]\(b = 22\)[/tex], and [tex]\(c = 7\)[/tex]. Let's break down the solution step-by-step.
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
[tex]\[
\text{Discriminant} = 22^2 - 4 \times 3 \times 7
\][/tex]
[tex]\[
\text{Discriminant} = 484 - 84 = 400
\][/tex]
2. Calculate the Roots:
Now, substitute the values into the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
[tex]\[
x = \frac{-22 \pm \sqrt{400}}{2 \times 3}
\][/tex]
[tex]\[
x = \frac{-22 \pm 20}{6}
\][/tex]
We have two cases to consider:
[tex]\[
x_1 = \frac{-22 + 20}{6} = \frac{-2}{6} = -\frac{1}{3}
\][/tex]
[tex]\[
x_2 = \frac{-22 - 20}{6} = \frac{-42}{6} = -7
\][/tex]
So, the roots of the equation are:
[tex]\[
x = -\frac{1}{3}, -7
\][/tex]
3. Check the Roots:
We can verify the roots by substituting them back into the original equation:
For [tex]\( x = -\frac{1}{3} \)[/tex]:
[tex]\[
3\left(-\frac{1}{3}\right)^2 + 22\left(-\frac{1}{3}\right) + 7
\][/tex]
[tex]\[
3 \times \frac{1}{9} - \frac{22}{3} + 7
\][/tex]
[tex]\[
\frac{3}{9} - \frac{22}{3} + 7 = \frac{1}{3} - \frac{22}{3} + 7
\][/tex]
[tex]\[
= \frac{1 - 22 + 21}{3} = \frac{0}{3} = 0
\][/tex]
For [tex]\( x = -7 \)[/tex]:
[tex]\[
3(-7)^2 + 22(-7) + 7
\][/tex]
[tex]\[
3 \times 49 - 154 + 7
\][/tex]
[tex]\[
147 - 154 + 7 = 0
\][/tex]
Both roots satisfy the original equation, confirming that the solutions are correct.
Thus, the solutions to the equation [tex]\( 3x^2 + 22x + 7 = 0 \)[/tex] are:
[tex]\[
\boxed{-\frac{1}{3}, -7}
\][/tex]
