To solve the quadratic equation [tex]\(u^2 - 2u - 24 = 0\)[/tex], we will use the quadratic formula. The quadratic formula for an equation of the form [tex]\(au^2 + bu + c = 0\)[/tex] is given by:
[tex]\[
u = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}
\][/tex]
Here, the coefficients are:
[tex]\[
a = 1, \quad b = -2, \quad c = -24
\][/tex]
1. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = (-2)^2 - 4(1)(-24)
\][/tex]
[tex]\[
\text{Discriminant} = 4 + 96
\][/tex]
[tex]\[
\text{Discriminant} = 100
\][/tex]
2. Since the discriminant is positive (100), there are two distinct real solutions.
3. Find the two solutions using the quadratic formula:
[tex]\[
u = \frac{{-b \pm \sqrt{\text{Discriminant}}}}{2a}
\][/tex]
Substituting the values:
[tex]\[
u = \frac{{-(-2) \pm \sqrt{100}}}{2(1)}
\][/tex]
[tex]\[
u = \frac{{2 \pm 10}}{2}
\][/tex]
- For the positive root:
[tex]\[
u_1 = \frac{{2 + 10}}{2}
\][/tex]
[tex]\[
u_1 = \frac{{12}}{2}
\][/tex]
[tex]\[
u_1 = 6
\][/tex]
- For the negative root:
[tex]\[
u_2 = \frac{{2 - 10}}{2}
\][/tex]
[tex]\[
u_2 = \frac{{-8}}{2}
\][/tex]
[tex]\[
u_2 = -4
\][/tex]
The solutions to the quadratic equation [tex]\(u^2 - 2u - 24 = 0\)[/tex] are:
[tex]\[
u = 6, -4
\][/tex]
