To find the zeros of the function [tex]\( f(a) = a^2 + 8a + 15 \)[/tex], we need to determine the values of [tex]\( a \)[/tex] for which [tex]\( f(a) = 0 \)[/tex]. This means solving the quadratic equation:
[tex]\[
a^2 + 8a + 15 = 0
\][/tex]
Here, the coefficients are [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 15 \)[/tex].
1. Calculate the Discriminant:
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting in our coefficients:
[tex]\[
\Delta = 8^2 - 4 \cdot 1 \cdot 15 = 64 - 60 = 4
\][/tex]
2. Determine the Roots Using the Quadratic Formula:
The quadratic formula for finding the roots of [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[
a = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in our values:
[tex]\[
a = \frac{-8 \pm \sqrt{4}}{2 \cdot 1} = \frac{-8 \pm 2}{2}
\][/tex]
3. Calculate the Two Possible Roots:
- For the positive square root:
[tex]\[
a_1 = \frac{-8 + 2}{2} = \frac{-6}{2} = -3
\][/tex]
- For the negative square root:
[tex]\[
a_2 = \frac{-8 - 2}{2} = \frac{-10}{2} = -5
\][/tex]
4. Conclusion:
Therefore, the zeros of the function [tex]\( f(a) = a^2 + 8a + 15 \)[/tex] are [tex]\( a = -3 \)[/tex] and [tex]\( a = -5 \)[/tex].
The roots can be written as:
[tex]\[
a_1 = -3 \quad \text{and} \quad a_2 = -5
\][/tex]
