To find the zeros of the function [tex]\( f(s) = s^2 + 12s + 32 \)[/tex], you need to determine the values of [tex]\( s \)[/tex] for which [tex]\( f(s) \)[/tex] equals zero. This involves solving the quadratic equation:
[tex]\[ s^2 + 12s + 32 = 0 \][/tex]
We can solve this quadratic equation using factoring. The standard form of a quadratic equation [tex]\( as^2 + bs + c \)[/tex] can often be factored into the form [tex]\( (s + p)(s + q) \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are numbers that need to be determined. These numbers should satisfy the conditions:
[tex]\[ p + q = -b \][/tex]
[tex]\[ pq = c \][/tex]
In this case, our equation is [tex]\( s^2 + 12s + 32 = 0 \)[/tex].
Here, [tex]\( b = 12 \)[/tex] and [tex]\( c = 32 \)[/tex]. We need to find two numbers [tex]\( p \)[/tex] and [tex]\( q \)[/tex] such that:
[tex]\[ p + q = -12 \][/tex]
[tex]\[ pq = 32 \][/tex]
Upon inspection, the numbers that satisfy these conditions are [tex]\( -8 \)[/tex] and [tex]\( -4 \)[/tex], because:
[tex]\[ (-8) + (-4) = -12 \][/tex]
[tex]\[ (-8) \times (-4) = 32 \][/tex]
Thus, the quadratic equation can be factored as:
[tex]\[ (s - 8)(s - 4) = 0 \][/tex]
Setting each factor equal to zero gives the solutions:
[tex]\[ s - 8 = 0 \quad \text{or} \quad s - 4 = 0 \][/tex]
So, the solutions are:
[tex]\[ s = -8 \][/tex]
[tex]\[ s = -4 \][/tex]
Therefore, the zeros of the function [tex]\( f(s) = s^2 + 12s + 32 \)[/tex] are [tex]\( s = -8 \)[/tex] and [tex]\( s = -4 \)[/tex].
