To find the zeros of the quadratic function [tex]\( f(x) = 4x^2 + 20x + 24 \)[/tex], we will use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are:
- [tex]\( a = 4 \)[/tex]
- [tex]\( b = 20 \)[/tex]
- [tex]\( c = 24 \)[/tex]
First, we calculate the discriminant, [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 20^2 - 4 \cdot 4 \cdot 24
\][/tex]
Calculate [tex]\( 20^2 \)[/tex]:
[tex]\[
20^2 = 400
\][/tex]
Calculate [tex]\( 4 \cdot 4 \cdot 24 \)[/tex]:
[tex]\[
4 \cdot 4 \cdot 24 = 384
\][/tex]
Now, subtract:
[tex]\[
\Delta = 400 - 384 = 16
\][/tex]
Next, we need to find the square root of the discriminant:
[tex]\[
\sqrt{\Delta} = \sqrt{16} = 4
\][/tex]
Now we can use the quadratic formula to find the two roots:
[tex]\[
x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Calculate the first root [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{-20 + 4}{2 \cdot 4} = \frac{-16}{8} = -2
\][/tex]
Calculate the second root [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-20 - 4}{2 \cdot 4} = \frac{-24}{8} = -3
\][/tex]
Thus, the zeros of the quadratic function [tex]\( f(x) = 4x^2 + 20x + 24 \)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad x = -3
\][/tex]
