To find the possible values of [tex]\( r \)[/tex] in the quadratic equation [tex]\( r^2 - 7r - 8 = 0 \)[/tex], we will solve it using the quadratic formula:
[tex]\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the quadratic equation is in the form [tex]\( ar^2 + br + c = 0 \)[/tex]. Identifying the coefficients, we have:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -7 \][/tex]
[tex]\[ c = -8 \][/tex]
First, calculate the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-7)^2 - 4(1)(-8) \][/tex]
[tex]\[ \Delta = 49 + 32 \][/tex]
[tex]\[ \Delta = 81 \][/tex]
With the discriminant calculated, we now find the roots of the equation using the quadratic formula:
[tex]\[ r = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Plugging in the values:
[tex]\[ r = \frac{-(-7) \pm \sqrt{81}}{2(1)} \][/tex]
[tex]\[ r = \frac{7 \pm 9}{2} \][/tex]
This gives us two possible solutions:
[tex]\[ r_1 = \frac{7 + 9}{2} = \frac{16}{2} = 8 \][/tex]
[tex]\[ r_2 = \frac{7 - 9}{2} = \frac{-2}{2} = -1 \][/tex]
Therefore, the possible values of [tex]\( r \)[/tex] are:
[tex]\[ r = 8 \][/tex]
[tex]\[ r = -1 \][/tex]
Thus, the correct answer is:
B. [tex]\( r = 8, r = -1 \)[/tex]
