Problem 50 - Algebra and Trigonometry

Question

Factor each perfect square trinomial. $$x^{2}+4 x+4$$

Step-by-Step Solution

Answer

The perfect square trinomial \(x^{2} + 4x + 4\) factors to \((x + 2)^2\).

Step 1 Identify the square terms

Firstly, identify the square terms in the trinomial. Here, \(x^{2}\) is the square of \(x\), and \(4\) is the square of \(2\). Therefore \(a = x\) and \(b = 2\).

Step 2 Verify double product term

Our next step will be to verify if the middle term in the trinomial is twice the product of the square root terms. Here \(2ab\) should equal \(4x\). If we multiply 2, \(a (x)\), and \(b (2)\), we get \(4x\), which is our middle term. This shows that the trinomial follows the pattern of a perfect square trinomial.

Step 3 Factorize the trinomial

With Step 1 and Step 2 confirmed, we can factor the trinomial using the identity \((a + b)^2\). Thus, the trinomial \(x^{2} + 4x + 4\) can be factored to become \((x + 2)^2\).