Factoring the Expression: 9x² + 12x + 4
To express the quadratic expression 9x² + 12x + 4 in factored form, we will follow these steps:
Step 1: Identify the Coefficients
In the expression ax² + bx + c, the coefficients are:
- a = 9
- b = 12
- c = 4
Step 2: Multiply a and c
Next, we multiply a and c: 9 * 4 = 36.
Step 3: Find Two Numbers that Multiply to ac and Add to b
We need to find two numbers that multiply to 36 and add up to 12. The numbers that fit are 6 and 6 since:
- 6 * 6 = 36
- 6 + 6 = 12
Step 4: Rewrite the Expression
We can rewrite the middle term (12x) using the two numbers we found:
9x² + 6x + 6x + 4
Step 5: Group the Terms
Now, we group the terms:
(9x² + 6x) + (6x + 4)
Step 6: Factor by Grouping
Next, we factor out the common factors from each group:
- From the first group (9x² + 6x), we can factor out 3x: 3x(3x + 2)
- From the second group (6x + 4), we can factor out 2: 2(3x + 2)
Putting this together, we have:
3x(3x + 2) + 2(3x + 2)
Step 7: Factor out the Common Binomial
Now we can see that (3x + 2) is a common factor:
(3x + 2)(3x + 2)
Final Result
Thus, the factored form of the expression 9x² + 12x + 4 is:
(3x + 2)²