Question
What is the factorization of 2x^2 + 7x + 6
Answers
GPT-4o mini
To factor the quadratic expression \(2x^2 + 7x + 6\), we can use the method of factoring by grouping. First, we need to find two numbers that multiply to the product of the leading coefficient (2) and the constant term (6), which is \(2 \times 6 = 12\), and that add up to the middle coefficient (7).
The two numbers that work are 3 and 4. Now, we can rewrite the middle term (7x) using these two numbers:
\[
2x^2 + 3x + 4x + 6
\]
Next, we group the terms:
\[
(2x^2 + 3x) + (4x + 6)
\]
Now, we can factor out the common factors in each group:
\[
x(2x + 3) + 2(2x + 3)
\]
Notice that \(2x + 3\) is a common factor:
\[
(x + 2)(2x + 3)
\]
Thus, the factorization of \(2x^2 + 7x + 6\) is:
\[
(2x + 3)(x + 2)
\]
The two numbers that work are 3 and 4. Now, we can rewrite the middle term (7x) using these two numbers:
\[
2x^2 + 3x + 4x + 6
\]
Next, we group the terms:
\[
(2x^2 + 3x) + (4x + 6)
\]
Now, we can factor out the common factors in each group:
\[
x(2x + 3) + 2(2x + 3)
\]
Notice that \(2x + 3\) is a common factor:
\[
(x + 2)(2x + 3)
\]
Thus, the factorization of \(2x^2 + 7x + 6\) is:
\[
(2x + 3)(x + 2)
\]