Let's solve for the missing root step-by-step given the function [tex]\( f(x) = x^2 - 2x - 3 \)[/tex] and the known root [tex]\( x = -1 \)[/tex].
This is a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex] where:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = -2 \][/tex]
[tex]\[ c = -3 \][/tex]
To find the roots of the quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-2)^2 - 4(1)(-3) \][/tex]
[tex]\[ \Delta = 4 + 12 \][/tex]
[tex]\[ \Delta = 16 \][/tex]
Now, the quadratic formula becomes:
[tex]\[ x = \frac{-(-2) \pm \sqrt{16}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{2 \pm 4}{2} \][/tex]
This gives us two solutions (roots):
[tex]\[ x_1 = \frac{2 + 4}{2} = \frac{6}{2} = 3 \][/tex]
[tex]\[ x_2 = \frac{2 - 4}{2} = \frac{-2}{2} = -1 \][/tex]
We are given one of the roots [tex]\( x = -1 \)[/tex], so the other root must be:
[tex]\[ x = 3 \][/tex]
Therefore, the missing root is:
[tex]\[ x = 3 \][/tex]
