Answer :
To find the zeros of the function [tex]\( f(x) = 1.1x^2 + 13x + 35.8 \)[/tex], we will use the quadratic formula. The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients from the quadratic function [tex]\( ax^2 + bx + c \)[/tex].
For our function, the coefficients are:
- [tex]\( a = 1.1 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 35.8 \)[/tex]
Step 1: Calculate the Discriminant
First, calculate the discriminant using the formula [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 = 13^2 = 169
\][/tex]
[tex]\[
4ac = 4 \times 1.1 \times 35.8 = 157.52
\][/tex]
[tex]\[
\text{Discriminant} = 169 - 157.52 = 11.48
\][/tex]
Since the discriminant is positive (11.48), the function has two real roots.
Step 2: Use the Quadratic Formula
Next, calculate the roots using the quadratic formula with the calculated discriminant:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Calculate each part:
1. Root 1:
[tex]\[
x_1 = \frac{-13 + \sqrt{11.48}}{2 \times 1.1}
\][/tex]
[tex]\[
x_1 = \frac{-13 + 3.387}{2.2}
\][/tex]
[tex]\[
x_1 \approx \frac{-9.613}{2.2}
\][/tex]
[tex]\[
x_1 \approx -4.369
\][/tex]
2. Root 2:
[tex]\[
x_2 = \frac{-13 - \sqrt{11.48}}{2 \times 1.1}
\][/tex]
[tex]\[
x_2 = \frac{-13 - 3.387}{2.2}
\][/tex]
[tex]\[
x_2 \approx \frac{-16.387}{2.2}
\][/tex]
[tex]\[
x_2 \approx -7.449
\][/tex]
Conclusion:
The zeros of the function [tex]\( f(x) = 1.1x^2 + 13x + 35.8 \)[/tex] are approximately [tex]\( x = -4.369 \)[/tex] and [tex]\( x = -7.449 \)[/tex].
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients from the quadratic function [tex]\( ax^2 + bx + c \)[/tex].
For our function, the coefficients are:
- [tex]\( a = 1.1 \)[/tex]
- [tex]\( b = 13 \)[/tex]
- [tex]\( c = 35.8 \)[/tex]
Step 1: Calculate the Discriminant
First, calculate the discriminant using the formula [tex]\( b^2 - 4ac \)[/tex]:
[tex]\[
b^2 = 13^2 = 169
\][/tex]
[tex]\[
4ac = 4 \times 1.1 \times 35.8 = 157.52
\][/tex]
[tex]\[
\text{Discriminant} = 169 - 157.52 = 11.48
\][/tex]
Since the discriminant is positive (11.48), the function has two real roots.
Step 2: Use the Quadratic Formula
Next, calculate the roots using the quadratic formula with the calculated discriminant:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Calculate each part:
1. Root 1:
[tex]\[
x_1 = \frac{-13 + \sqrt{11.48}}{2 \times 1.1}
\][/tex]
[tex]\[
x_1 = \frac{-13 + 3.387}{2.2}
\][/tex]
[tex]\[
x_1 \approx \frac{-9.613}{2.2}
\][/tex]
[tex]\[
x_1 \approx -4.369
\][/tex]
2. Root 2:
[tex]\[
x_2 = \frac{-13 - \sqrt{11.48}}{2 \times 1.1}
\][/tex]
[tex]\[
x_2 = \frac{-13 - 3.387}{2.2}
\][/tex]
[tex]\[
x_2 \approx \frac{-16.387}{2.2}
\][/tex]
[tex]\[
x_2 \approx -7.449
\][/tex]
Conclusion:
The zeros of the function [tex]\( f(x) = 1.1x^2 + 13x + 35.8 \)[/tex] are approximately [tex]\( x = -4.369 \)[/tex] and [tex]\( x = -7.449 \)[/tex].
