Problem 53 - Algebra and Trigonometry

Question

Factor each perfect square trinomial. $$4 x^{2}+4 x+1$$

Step-by-Step Solution

Answer

The factorization of the given perfect square trinomial \(4x^{2} + 4x + 1\) is \((2x+1)^{2}\).

Step 1: Identify 'a' and 'b'

In the given trinomial \(4x^{2} + 4x + 1\), 'a' corresponds to the square root of the coefficient of the first term, hence \(a=2x\), and 'b' corresponds to the square root of the third term, hence \(b=1\).

Step 2: Use the Perfect Square Trinomial Formula

The formula for a perfect square trinomial is \((a+b)^{2}=a^{2} + 2ab + b^{2}\) or \((a-b)^{2}=a^{2} - 2ab + b^{2}\). In our case, the trinomial corresponds to the positive version, thus we employ \((a+b)^{2}=a^{2} + 2ab + b^{2}\). Substitute the identified values of 'a' and 'b' into this formula.

Step 3: Simplify and Factorise

Substituting \(a=2x\) and \(b=1\) into the formula, we get \((2x+1)^{2}\) as the factorization of the given trinomial.