To solve the quadratic equation [tex]\(3x^2 + 12x + 6 = 0\)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\( a = 3 \)[/tex], [tex]\( b = 12 \)[/tex], and [tex]\( c = 6 \)[/tex].
Step-by-Step Solution:
1. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 12^2 - 4 \cdot 3 \cdot 6 \][/tex]
[tex]\[ \Delta = 144 - 72 \][/tex]
[tex]\[ \Delta = 72 \][/tex]
2. Compute the solutions using the quadratic formula:
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Substituting the discriminant ([tex]\(\Delta = 72\)[/tex]) and the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ x = \frac{-12 \pm \sqrt{72}}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-12 \pm \sqrt{72}}{6} \][/tex]
3. Simplify the expression:
First, simplify [tex]\(\sqrt{72}\)[/tex]:
[tex]\[ \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \][/tex]
Now substitute back:
[tex]\[ x = \frac{-12 \pm 6\sqrt{2}}{6} \][/tex]
Simplify the fraction:
[tex]\[ x = -2 \pm \sqrt{2} \][/tex]
Thus, the solutions to the quadratic equation [tex]\(3x^2 + 12x + 6 = 0\)[/tex] are:
[tex]\[ x = -2 \pm \sqrt{2} \][/tex]
The correct answer is:
A) [tex]\( x = -2 \pm \sqrt{2} \)[/tex]
