To solve the quadratic equation [tex]\(3w^2 + 14 = -13w\)[/tex] for [tex]\(w\)[/tex], follow these steps:
1. Rewrite the equation in standard form:
Begin by bringing all terms to one side of the equation so it is set to zero. This means you need to add [tex]\(13w\)[/tex] to both sides of the equation.
[tex]\[
3w^2 + 14 + 13w = 0
\][/tex]
Rearrange the terms to match the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[
3w^2 + 13w + 14 = 0
\][/tex]
2. Identify the coefficients:
In the standard form [tex]\(3w^2 + 13w + 14 = 0\)[/tex], we can see that:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 13\)[/tex]
- [tex]\(c = 14\)[/tex]
3. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
w = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
w = \frac{{-13 \pm \sqrt{{13^2 - 4 \cdot 3 \cdot 14}}}}{2 \cdot 3}
\][/tex]
4. Simplify under the square root:
Compute the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex]:
[tex]\[
\Delta = 13^2 - 4 \cdot 3 \cdot 14
\][/tex]
Calculate [tex]\(13^2 = 169\)[/tex] and [tex]\(4 \cdot 3 \cdot 14 = 168\)[/tex]:
[tex]\[
\Delta = 169 - 168 = 1
\][/tex]
5. Substitute back into the quadratic formula:
Now we have:
[tex]\[
w = \frac{{-13 \pm \sqrt{1}}}{6}
\][/tex]
[tex]\(\sqrt{1} = 1\)[/tex], so:
[tex]\[
w = \frac{{-13 \pm 1}}{6}
\][/tex]
6. Find the two solutions:
[tex]\[
w_1 = \frac{{-13 + 1}}{6} = \frac{{-12}}{6} = -2
\][/tex]
[tex]\[
w_2 = \frac{{-13 - 1}}{6} = \frac{{-14}}{6} = -\frac{14}{6} = -\frac{7}{3}
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(3w^2 + 14 + 13w = 0\)[/tex] are:
[tex]\[
w = -2 \quad \text{and} \quad w = -\frac{7}{3}
\][/tex]
