Select the correct answer.

Solve the following equation for [tex]x[/tex]:
[tex]\[ x^2 - 9x + 18 = 0 \][/tex]

A. [tex]x = -3 ; x = 6[/tex]
B. [tex]x = 3 ; x = 6[/tex]
C. [tex]x = -3 ; x = -6[/tex]
D. [tex]x = 3 ; x = -6[/tex]



Answer :

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]