To solve the quadratic equation [tex]\( m^2 + 8m + 12 = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[ m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients of the quadratic equation [tex]\( m^2 + 8m + 12 = 0 \)[/tex] are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 8\)[/tex]
- [tex]\(c = 12\)[/tex]
Let's go through the steps to find the roots of the equation.
1. Calculate the discriminant:
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[\Delta = b^2 - 4ac\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (8)^2 - 4(1)(12) = 64 - 48 = 16
\][/tex]
2. Determine the nature of the roots:
The discriminant is 16, which is greater than 0. This means the quadratic equation has two distinct real roots.
3. Calculate the roots using the quadratic formula:
Let's calculate the two roots [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex]:
[tex]\[
m_1 = \frac{-b + \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
m_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[
m_1 = \frac{-(8) + \sqrt{16}}{2(1)} = \frac{-8 + 4}{2} = \frac{-4}{2} = -2
\][/tex]
[tex]\[
m_2 = \frac{-(8) - \sqrt{16}}{2(1)} = \frac{-8 - 4}{2} = \frac{-12}{2} = -6
\][/tex]
So, the solutions to the quadratic equation [tex]\( m^2 + 8m + 12 = 0 \)[/tex] are:
[tex]\[
m_1 = -2 \quad \text{and} \quad m_2 = -6
\][/tex]
Thus, the roots of the equation are [tex]\( m = -2 \)[/tex] and [tex]\( m = -6 \)[/tex].
