To find the zeros of the quadratic function [tex]\( f(x) = x^2 + x - 12 \)[/tex], we need to solve the equation [tex]\( x^2 + x - 12 = 0 \)[/tex].
For any quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
1. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -12 \)[/tex].
2. The quadratic formula is used to find the solutions (zeros) of the quadratic equation:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
3. Substitute the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(1) \pm \sqrt{(1)^2 - 4(1)(-12)}}{2(1)}
\][/tex]
4. Simplify the expression inside the square root (the discriminant):
[tex]\[
\Delta = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-12) = 1 + 48 = 49
\][/tex]
5. Take the square root of the discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
6. Substitute back into the quadratic formula:
[tex]\[
x = \frac{-1 \pm 7}{2}
\][/tex]
7. Calculate the two possible values for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-1 + 7}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
x = \frac{-1 - 7}{2} = \frac{-8}{2} = -4
\][/tex]
Thus, the zeros of the function [tex]\( f(x) = x^2 + x - 12 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -4 \)[/tex].
Based on the given choices, the correct answer is:
A. [tex]\( x = -4 \)[/tex] and [tex]\( x = 3 \)[/tex]
