To find the solutions for the quadratic equation [tex]\(3x^2 + 8x + 4 = 0\)[/tex], we need to determine the roots of the equation. This is typically done using 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 terms in the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. For our specific equation [tex]\(3x^2 + 8x + 4 = 0\)[/tex], the coefficients are:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 8\)[/tex]
- [tex]\(c = 4\)[/tex]
Given the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we substitute them into the quadratic formula to solve for [tex]\(x\)[/tex]:
1. Compute the discriminant: [tex]\(b^2 - 4ac\)[/tex]
[tex]\[
b^2 - 4ac = 8^2 - 4 \cdot 3 \cdot 4 = 64 - 48 = 16
\][/tex]
2. Compute the two roots using the quadratic formula:
[tex]\[
x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} = \frac{-8 + \sqrt{16}}{2 \cdot 3} = \frac{-8 + 4}{6} = \frac{-4}{6} = -\frac{2}{3}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} = \frac{-8 - \sqrt{16}}{2 \cdot 3} = \frac{-8 - 4}{6} = \frac{-12}{6} = -2
\][/tex]
These calculations yield the roots of the quadratic equation. Therefore, the solution set for the equation [tex]\(3x^2 + 8x + 4 = 0\)[/tex] is:
[tex]\[
\boxed{-\frac{2}{3}, -2}
\][/tex]
In decimal form, these roots are approximately:
[tex]\[
\boxed{-0.66666667, -2}
\][/tex]
Hence, the values which satisfy the given quadratic equation are [tex]\(-0.66666667\)[/tex] and [tex]\(-2\)[/tex].
