To split the middle term of the quadratic expression 3a^2 + 8ab + 4b^2, we need to find two terms that add up to 8ab, the coefficient of the middle term, and then rewrite the expression as a sum of these two terms. The technique we use is called "completing the square."
The expression is: 3a^2 + 8ab + 4b^2.
Step 1: Look for two terms whose product is the product of the coefficients of the first and last terms. The product of 3a^2 and 4b^2 is 12a^2b^2.
Step 2: Split the middle term (8ab) into two terms using the product obtained in step 1. The goal is to rewrite 8ab as the sum of two terms whose coefficients multiply to give us 12a^2b^2.
Let's find two numbers whose product is 12a^2b^2 and whose sum is 8ab:
The numbers are 6ab and 2ab because:
6ab * 2ab = 12a^2b^2 and 6ab + 2ab = 8ab.
Step 3: Now, rewrite the middle term using the two terms obtained in step 2:
3a^2 + 6ab + 2ab + 4b^2.
Step 4: Group the terms in pairs:
(3a^2 + 6ab) + (2ab + 4b^2).
Step 5: Factor out the common terms from each pair:
3a(a + 2b) + 2b(a + 2b).
Step 6: Now, notice that we have a common factor of (a + 2b) in both terms. Factor it out:
(a + 2b)(3a + 2b).
So, after splitting the middle term, the original expression can be written as (a + 2b)(3a + 2b).
Answers & Comments
Answer:
To split the middle term of the quadratic expression 3a^2 + 8ab + 4b^2, we need to find two terms that add up to 8ab, the coefficient of the middle term, and then rewrite the expression as a sum of these two terms. The technique we use is called "completing the square."
The expression is: 3a^2 + 8ab + 4b^2.
Step 1: Look for two terms whose product is the product of the coefficients of the first and last terms. The product of 3a^2 and 4b^2 is 12a^2b^2.
Step 2: Split the middle term (8ab) into two terms using the product obtained in step 1. The goal is to rewrite 8ab as the sum of two terms whose coefficients multiply to give us 12a^2b^2.
Let's find two numbers whose product is 12a^2b^2 and whose sum is 8ab:
The numbers are 6ab and 2ab because:
6ab * 2ab = 12a^2b^2 and 6ab + 2ab = 8ab.
Step 3: Now, rewrite the middle term using the two terms obtained in step 2:
3a^2 + 6ab + 2ab + 4b^2.
Step 4: Group the terms in pairs:
(3a^2 + 6ab) + (2ab + 4b^2).
Step 5: Factor out the common terms from each pair:
3a(a + 2b) + 2b(a + 2b).
Step 6: Now, notice that we have a common factor of (a + 2b) in both terms. Factor it out:
(a + 2b)(3a + 2b).
So, after splitting the middle term, the original expression can be written as (a + 2b)(3a + 2b).
[tex] \tt{ {3a}^{2} + 8ab + {4b}^{2} }[/tex]
[tex] \tt{ = {3a}^{2} + 6ab + 2ab + {4b}^{2} }[/tex]
[tex] \tt{ = 3a(a + 2b) + 2b(a + 2b}[/tex]
[tex] \tt = \red{(3a + 2b)(a + 2b)}[/tex]