To solve the quadratic equation [tex]\( x^2 - 9x + 18 = 0 \)[/tex], we'll use the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation [tex]\( x^2 - 9x + 18 = 0 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -9 \)[/tex]
- [tex]\( c = 18 \)[/tex]
First, we calculate the discriminant ([tex]\( \Delta \)[/tex]) using the formula [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[
\Delta = (-9)^2 - 4 \cdot 1 \cdot 18
\][/tex]
[tex]\[
\Delta = 81 - 72
\][/tex]
[tex]\[
\Delta = 9
\][/tex]
Since the discriminant is positive ([tex]\( \Delta > 0 \)[/tex]), we have two real solutions. We now substitute into the quadratic formula to find the roots of the equation.
The quadratic formula states:
[tex]\[
x = \frac{-(-9) \pm \sqrt{9}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{9 \pm 3}{2}
\][/tex]
This gives us two solutions:
1. For the positive root:
[tex]\[
x_1 = \frac{9 + 3}{2}
\][/tex]
[tex]\[
x_1 = \frac{12}{2}
\][/tex]
[tex]\[
x_1 = 6
\][/tex]
2. For the negative root:
[tex]\[
x_2 = \frac{9 - 3}{2}
\][/tex]
[tex]\[
x_2 = \frac{6}{2}
\][/tex]
[tex]\[
x_2 = 3
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( x^2 - 9x + 18 = 0 \)[/tex] are: [tex]\( x = 6 \)[/tex] and [tex]\( x = 3 \)[/tex].
The correct answer is:
B. [tex]\( x = 3 ; x = 6 \)[/tex]
