To solve the equation
[tex]\[
3x(x+6) = -10,
\][/tex]
we follow these steps:
1. Distribute the [tex]\(3x\)[/tex] through the parentheses:
[tex]\[
3x^2 + 18x = -10.
\][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[
3x^2 + 18x + 10 = 0.
\][/tex]
3. Now, we need to solve the quadratic equation [tex]\(3x^2 + 18x + 10 = 0\)[/tex]. To find the roots of the quadratic equation, we can use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
\][/tex]
In our equation, [tex]\(a = 3\)[/tex], [tex]\(b = 18\)[/tex], and [tex]\(c = 10\)[/tex]. Plugging these values into the quadratic formula:
[tex]\[
x = \frac{-18 \pm \sqrt{18^2 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}.
\][/tex]
4. First, calculate the discriminant:
[tex]\[
18^2 - 4 \cdot 3 \cdot 10 = 324 - 120 = 204.
\][/tex]
5. Then, take the square root of the discriminant:
[tex]\[
\sqrt{204}.
\][/tex]
6. Substitute back into the quadratic formula:
[tex]\[
x = \frac{-18 \pm \sqrt{204}}{6}.
\][/tex]
7. Simplify the square root if possible:
[tex]\[
\sqrt{204} = 2\sqrt{51},
\][/tex]
so the formula becomes:
[tex]\[
x = \frac{-18 \pm 2\sqrt{51}}{6}.
\][/tex]
8. Simplify the fractions:
[tex]\[
x = \frac{-18}{6} \pm \frac{2\sqrt{51}}{6} = -3 \pm \frac{\sqrt{51}}{3}.
\][/tex]
Thus, the solutions to the equation are:
[tex]\[
x = -3 \pm \frac{\sqrt{51}}{3}.
\][/tex]
