To find the zeros of the quadratic function [tex]\( f(x) = x^2 + x - 20 \)[/tex], we need to solve the equation [tex]\( x^2 + x - 20 = 0 \)[/tex].
Let's solve this step-by-step by factoring the quadratic equation:
1. Write the equation: [tex]\( x^2 + x - 20 = 0 \)[/tex].
2. We need to find two numbers that multiply to give [tex]\(-20\)[/tex] (the constant term) and add to give [tex]\(1\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
3. The two numbers that satisfy these conditions are [tex]\(5\)[/tex] and [tex]\(-4\)[/tex] because:
- [tex]\( 5 \times (-4) = -20 \)[/tex]
- [tex]\( 5 + (-4) = 1 \)[/tex]
4. Using these numbers, we can express the quadratic equation as:
[tex]\[
x^2 + x - 20 = (x + 5)(x - 4) = 0
\][/tex]
5. To find the zeros, we set each factor equal to zero:
[tex]\[
x + 5 = 0 \quad \text{or} \quad x - 4 = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = -5 \quad \text{or} \quad x = 4
\][/tex]
So, the zeros of the function [tex]\( f(x) = x^2 + x - 20 \)[/tex] are [tex]\( x = -5 \)[/tex] and [tex]\( x = 4 \)[/tex].
Among the given choices, the correct answer is:
D. [tex]\( x = -5 \)[/tex] and [tex]\( x = 4 \)[/tex]
