Factor by using the​ perfect-square trinomial formula.
9 x squared plus 12 x plus 4


To factor the expression \(9x^2 + 12x + 4\) using the perfect-square trinomial formula, follow these steps:

### Step 1: Recognize the Form of a Perfect-Square Trinomial
A perfect-square trinomial takes the form:
\[
a^2 + 2ab + b^2 = (a + b)^2
\]
where \(a^2\) and \(b^2\) are perfect squares, and \(2ab\) is the middle term.

### Step 2: Identify \(a\) and \(b\)
In the trinomial \(9x^2 + 12x + 4\):
- The first term \(9x^2\) is a perfect square. Here, \(a = 3x\) because \((3x)^2 = 9x^2\).
- The last term \(4\) is also a perfect square. Here, \(b = 2\) because \(2^2 = 4\).

### Step 3: Verify the Middle Term
The middle term should be \(2ab\). Let's check:
\[
2ab = 2 \times 3x \times 2 = 12x
\]
This matches the middle term of the given trinomial, confirming it is a perfect-square trinomial.

### Step 4: Write the Factored Form
Since the trinomial fits the perfect-square trinomial form, you can factor it as:
\[
(3x + 2)^2
\]

### Final Answer
The factored form of \(9x^2 + 12x + 4\) is \((3x + 2)^2\).

Would you like any more details or have any questions?

Here are 5 related questions you might find useful:

1. How do you factor other perfect-square trinomials with different coefficients?
2. Can you explain the process for factoring non-perfect-square trinomials?
3. What is the difference between factoring and expanding an expression?
4. How would you factor the trinomial if it had a negative middle term?
5. Can the perfect-square trinomial formula be applied to any trinomial?

**Tip:** Always verify the middle term when factoring a trinomial using the perfect-square formula to ensure the correct factorization.

Learn how to factor the perfect-square trinomial 9x^2 + 12x + 4 using the perfect-square trinomial formula. Follow a detailed step-by-step solution to factor this algebraic expression, suitable for high school students (Grades 9-12).

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