To solve the quadratic equation [tex]\( x^2 - 2x - 15 = 0 \)[/tex] and find the other solution given that one of the solutions is [tex]\( x = -3 \)[/tex], we can use the following steps:
1. Understanding the Equation:
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. For the equation given:
[tex]\[
x^2 - 2x - 15 = 0
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -15 \)[/tex].
2. Using the Known Solution:
We know that one of the solutions is [tex]\( x = -3 \)[/tex].
3. Quadratic Form:
From the quadratic formula, the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] are:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
Substituting [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = -15 \)[/tex]:
[tex]\[
x = \frac{{-(-2) \pm \sqrt{{(-2)^2 - 4 \cdot 1 \cdot (-15)}}}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{{2 \pm \sqrt{{4 + 60}}}}{2}
\][/tex]
[tex]\[
x = \frac{{2 \pm \sqrt{{64}}}}{2}
\][/tex]
[tex]\[
x = \frac{{2 \pm 8}}{2}
\][/tex]
This gives us two potential roots:
[tex]\[
x = \frac{{2 + 8}}{2} = \frac{10}{2} = 5
\][/tex]
[tex]\[
x = \frac{{2 - 8}}{2} = \frac{-6}{2} = -3
\][/tex]
4. Identifying the Roots:
We already know that one of the roots is [tex]\( x = -3 \)[/tex]. Therefore, the other root must be:
[tex]\[
x = 5
\][/tex]
Thus, the other solution to the quadratic equation [tex]\( x^2 - 2x - 15 = 0 \)[/tex] is:
[tex]\[ x = 5 \][/tex]
So, the correct answer is:
[tex]\[ x = 5 \][/tex]
