Question

fill in the blank 2.5 points
factor to find the zeros of the function:
$6x^2 + 72x + 192 = 0$
answers should be written as x=#,# with no spaces. write fractions as #/#.
for example, if your answer was x = 2 and x = -3/4, you would write your answer as x=2,3/4
type your answer...

Explanation:

Step1: Factor out the GCF

First, we find the greatest common factor (GCF) of the terms \(6x^{2}\), \(72x\), and \(192\). The GCF of \(6\), \(72\), and \(192\) is \(6\). So we factor out \(6\) from the left - hand side of the equation:
\(6(x^{2}+12x + 32)=0\)

Step2: Factor the quadratic

Now we need to factor the quadratic expression \(x^{2}+12x + 32\). We need to find two numbers that multiply to \(32\) and add up to \(12\). The numbers \(8\) and \(4\) satisfy this condition because \(8\times4 = 32\) and \(8 + 4=12\). So we can factor \(x^{2}+12x + 32\) as \((x + 8)(x+4)\).
The equation becomes \(6(x + 8)(x + 4)=0\)

Step3: Find the zeros

To find the zeros of the function, we set each factor equal to zero (since if a product of factors is zero, then at least one of the factors must be zero).

• For the factor \(6\), it is never zero for any real value of \(x\).
• For the factor \((x + 8)\):

Set \(x+8 = 0\), then solve for \(x\) by subtracting \(8\) from both sides. We get \(x=-8\).

• For the factor \((x + 4)\):

Set \(x + 4=0\), then solve for \(x\) by subtracting \(4\) from both sides. We get \(x=-4\).

Answer:

\(x=-8,-4\)