To solve the quadratic equation [tex]\( 3n^2 - 60n + 192 = 0 \)[/tex] by factoring, we follow a systematic step-by-step procedure:
### Step 1: Write the equation in standard form
The given equation is already in standard form:
[tex]\[ 3n^2 - 60n + 192 = 0 \][/tex]
### Step 2: Factor out the greatest common factor (GCF), if any
First, check for any common factors among the coefficients. In this case, there is no GCF other than 1, so we proceed directly to factoring the quadratic trinomial.
### Step 3: Factor the quadratic trinomial
We need to factor the quadratic expression [tex]\( 3n^2 - 60n + 192 \)[/tex]. Factoring by grouping or applying the quadratic formula can help, but since the original problem suggests that factored form exists, we shall follow that route.
### Step 4: Checking for correct factors
Based on the roots (zeros) given, we need factors of [tex]\( 3n^2 - 60n + 192 \)[/tex] equating to zero.
We know the solutions are [tex]\( n = 4 \)[/tex] and [tex]\( n = 16 \)[/tex]. Therefore, we can express the trinomial as a product of two binomials:
[tex]\[ (n - 4) \quad \text{and} \quad (n - 16) \][/tex]
Let's form the quadratic equation using these solutions:
[tex]\[ 0 = 3 (n - 4)(n - 16) \][/tex]
### Step 5: Verify and write the solutions
Now, let’s quickly expand to confirm our factored form:
[tex]\[ 3(n - 4)(n - 16) \][/tex]
[tex]\[ = 3[n^2 - 16n - 4n + 64] \][/tex]
[tex]\[ = 3[n^2 - 20n + 64] \][/tex]
[tex]\[ = 3n^2 - 60n + 192 \][/tex]
We see it returns to the original equation, which confirms the factorization is correct.
Hence the solutions for the quadratic equation based on its roots are:
[tex]\[ n = 4, 16 \][/tex]
### Conclusion
The correct choice is:
A. The solution is [tex]\( n = 4, 16 \)[/tex]
Thus, the filled in answer box should be:
[tex]\[ n = 4, 16 \][/tex]
