the diagram represents the factorization of ( a^{2}+8a + 12 ). what is the missing number that will complete…
the diagram represents the factorization of ( a^{2}+8a + 12 ). what is the missing number that will complete the factorization? 6 8 12 24
Answer
Explanation:
Step1: Analyze the factorization pattern
For a quadratic (x^{2}+bx + c=(x + m)(x + n)=x^{2}+(m + n)x+mn). Here (x = a), (b = 8), (c = 12). We know one of the factors is ((a + 2)) (from the left - hand side of the diagram where we have (2) in the first column, third row). Let the other factor be ((a + k)). Then (a^{2}+8a + 12=(a + 2)(a + k)=a^{2}+(2 + k)a+2k).
Step2: Use the coefficient of the linear term
We know that the coefficient of (a) is (8). So (2 + k=8) (comparing the coefficients of (a) in (a^{2}+(2 + k)a+2k) and (a^{2}+8a + 12)). Solving for (k), we get (k=8 - 2=6). Also, checking the constant term: if (k = 6), then (2k=2\times6 = 12) (which matches the constant term in (a^{2}+8a + 12)). Another way is to look at the area - model (diagram). The product of the numbers in the first column ((a) and (2)) and the second column ((a) and (?)) gives the terms of the quadratic. The term (6a) is obtained as (a\times6).
Answer:
6