To solve the quadratic equation [tex]\( x^2 - 9x + 20 = 0 \)[/tex], we can use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this equation, [tex]\( a = 1 \)[/tex], [tex]\( b = -9 \)[/tex], and [tex]\( c = 20 \)[/tex].
1. First, we calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting 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]
2. Next, we find the two solutions using the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{\Delta}}}{2a} \][/tex]
Substituting [tex]\( b = -9 \)[/tex], [tex]\( a = 1 \)[/tex], and [tex]\( \Delta = 1 \)[/tex]:
[tex]\[ x_1 = \frac{{-(-9) + \sqrt{1}}}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{{9 + 1}}{2} \][/tex]
[tex]\[ x_1 = \frac{10}{2} \][/tex]
[tex]\[ x_1 = 5 \][/tex]
[tex]\[ x_2 = \frac{{-(-9) - \sqrt{1}}}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{{9 - 1}}{2} \][/tex]
[tex]\[ x_2 = \frac{8}{2} \][/tex]
[tex]\[ x_2 = 4 \][/tex]
Therefore, the solutions to the equation [tex]\( x^2 - 9x + 20 = 0 \)[/tex] are [tex]\( x = 5 \)[/tex] and [tex]\( x = 4 \)[/tex].
The correct answer is:
B. [tex]\( x = 4 ; x = 5 \)[/tex]
