To solve the quadratic equation [tex]\(3x^2 + 18x + 27 = 0\)[/tex], we follow these steps:
1. Identify the coefficients:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 18\)[/tex]
- [tex]\(c = 27\)[/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]:
[tex]\[
\Delta = 18^2 - 4 \cdot 3 \cdot 27
\][/tex]
[tex]\[
\Delta = 324 - 324
\][/tex]
[tex]\[
\Delta = 0
\][/tex]
3. Interpret the discriminant:
The discriminant being zero ([tex]\(\Delta = 0\)[/tex]) indicates that the quadratic equation has exactly one real solution, also known as a repeated root.
4. Use the quadratic formula:
The solutions of the quadratic equation are given by:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Since [tex]\(\Delta = 0\)[/tex], this simplifies to:
[tex]\[
x = \frac{-b}{2a}
\][/tex]
5. Calculate the solution:
Substituting the values [tex]\(b = 18\)[/tex] and [tex]\(a = 3\)[/tex]:
[tex]\[
x = \frac{-18}{2 \cdot 3}
\][/tex]
[tex]\[
x = \frac{-18}{6}
\][/tex]
[tex]\[
x = -3
\][/tex]
Therefore, the solution to the equation [tex]\(3x^2 + 18x + 27 = 0\)[/tex] is:
[tex]\[
x = -3
\][/tex]
Since the discriminant is zero, this means we have a repeated root, so:
[tex]\[
x_1 = x_2 = -3
\][/tex]
