Let's find the solutions for the quadratic equation [tex]\(x^2 - 9x + 20 = 0\)[/tex].
This is in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -9\)[/tex]
- [tex]\(c = 20\)[/tex]
To solve this, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, calculate the discriminant [tex]\(\Delta\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-9)^2 - 4 \cdot 1 \cdot 20 \][/tex]
[tex]\[ \Delta = 81 - 80 \][/tex]
[tex]\[ \Delta = 1 \][/tex]
Now, use the quadratic formula with the calculated discriminant [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x = \frac{-(-9) \pm \sqrt{1}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{9 \pm 1}{2} \][/tex]
Hence, the two possible solutions for [tex]\(x\)[/tex] are:
[tex]\[ x_1 = \frac{9 + 1}{2} = \frac{10}{2} = 5 \][/tex]
[tex]\[ x_2 = \frac{9 - 1}{2} = \frac{8}{2} = 4 \][/tex]
Therefore, the solutions are [tex]\(x = 5\)[/tex] and [tex]\(x = 4\)[/tex].
The correct answer is:
B. [tex]\( x = 4 ; x = 5 \)[/tex]
