To solve the equation [tex]\(3x(x+6) = -10\)[/tex] using the most direct method, follow these steps:
1. First, expand and rearrange the equation:
[tex]\[
3x^2 + 18x = -10
\][/tex]
2. Move all terms to one side to set the equation to zero:
[tex]\[
3x^2 + 18x + 10 = 0
\][/tex]
3. Now, use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex] to find the solutions of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. Here, [tex]\(a = 3\)[/tex], [tex]\(b = 18\)[/tex], and [tex]\(c = 10\)[/tex].
4. Calculate the discriminant [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[
\Delta = 18^2 - 4 \cdot 3 \cdot 10 = 324 - 120 = 204
\][/tex]
5. Substitute back into the quadratic formula:
[tex]\[
x = \frac{-18 \pm \sqrt{204}}{2 \cdot 3}
\][/tex]
6. Simplify the expression under the square root and the resulting fractions:
[tex]\[
x = \frac{-18 \pm \sqrt{204}}{6}
\][/tex]
7. Further simplify [tex]\(\sqrt{204}\)[/tex]. We find that:
[tex]\[
\sqrt{204} = \sqrt{4 \cdot 51} = 2\sqrt{51}
\][/tex]
8. Substitute this back into our expression for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-18 \pm 2\sqrt{51}}{6}
\][/tex]
9. Simplify the fractions by dividing by 2:
[tex]\[
x = \frac{-9 \pm \sqrt{51}}{3}
\][/tex]
Therefore, the solution to the equation [tex]\(3x(x + 6) = -10\)[/tex] in the exact, most simplified form is:
[tex]\[
x = -3 \pm \frac{\sqrt{51}}{3}
\][/tex]
