Sure! Let's solve the quadratic equation [tex]\(6x^2 + 36x + 54 = 0\)[/tex] step by step.
1. Identify the coefficients:
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In our equation, the coefficients are:
[tex]\[
a = 6, \quad b = 36, \quad c = 54
\][/tex]
2. 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]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], we get:
[tex]\[
\Delta = 36^2 - 4 \cdot 6 \cdot 54 = 1296 - 1296 = 0
\][/tex]
3. Solve using the quadratic formula:
The solutions for the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since the discriminant [tex]\(\Delta\)[/tex] is 0, we have:
[tex]\[
x = \frac{-b \pm \sqrt{0}}{2a} = \frac{-b}{2a}
\][/tex]
4. Calculate the solution:
Substitute [tex]\(a = 6\)[/tex] and [tex]\(b = 36\)[/tex] into the formula:
[tex]\[
x = \frac{-36}{2 \cdot 6} = \frac{-36}{12} = -3
\][/tex]
Since the discriminant is 0, there is only one unique solution for this quadratic equation. Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[
x = -3
\][/tex]
