To solve the quadratic equation [tex]\(3x^2 + x - 14 = 0\)[/tex], follow these steps:
1. Identify the coefficients:
In the standard form of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex], the coefficients are:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = -14\)[/tex]
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 3\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -14\)[/tex].
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = 1^2 - 4 \cdot 3 \cdot (-14)
\][/tex]
Simplifying the expression in the discriminant:
[tex]\[
\Delta = 1 + 168 = 169
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Substitute [tex]\(\Delta\)[/tex] back into the quadratic formula:
[tex]\[
x = \frac{-1 \pm \sqrt{169}}{2 \cdot 3}
\][/tex]
Simplify further by calculating the square root of the discriminant:
[tex]\[
\sqrt{169} = 13
\][/tex]
Now the equation becomes:
[tex]\[
x = \frac{-1 \pm 13}{6}
\][/tex]
5. Find the roots:
This gives us two possible solutions:
[tex]\[
x = \frac{-1 + 13}{6} \quad \text{and} \quad x = \frac{-1 - 13}{6}
\][/tex]
Simplifying these:
[tex]\[
x = \frac{12}{6} = 2
\][/tex]
[tex]\[
x = \frac{-14}{6} = -\frac{7}{3}
\][/tex]
Therefore, the solutions to the equation [tex]\(3x^2 + x - 14 = 0\)[/tex] are:
[tex]\[
x = 2 \quad \text{and} \quad x = -\frac{7}{3}
\][/tex]
