To find the roots of the quadratic equation [tex]\(x^2 - 2x - 15 = 0\)[/tex], we can use the quadratic formula, which is given by:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
For the equation [tex]\( x^2 - 2x - 15 = 0 \)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(c = -15\)[/tex]
Step 1: Calculate the discriminant [tex]\(D\)[/tex]:
[tex]\[ D = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ D = (-2)^2 - 4(1)(-15) \][/tex]
[tex]\[ D = 4 + 60 \][/tex]
[tex]\[ D = 64 \][/tex]
Step 2: Use the quadratic formula to find the roots. There are two roots, depending on the sign taken in the formula:
[tex]\[ x = \frac{-(-2) \pm \sqrt{64}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm 8}{2} \][/tex]
Step 3: Calculate each root separately.
For the positive case [tex]\((+)\)[/tex]:
[tex]\[ x = \frac{2 + 8}{2} \][/tex]
[tex]\[ x = \frac{10}{2} \][/tex]
[tex]\[ x = 5 \][/tex]
For the negative case [tex]\((-)\)[/tex]:
[tex]\[ x = \frac{2 - 8}{2} \][/tex]
[tex]\[ x = \frac{-6}{2} \][/tex]
[tex]\[ x = -3 \][/tex]
So, the roots of the quadratic equation [tex]\( x^2 - 2x - 15 = 0 \)[/tex] are:
[tex]\[
x = 5, -3
\][/tex]
