Problem 56 - Algebra and Trigonometry

Question

Factor each perfect square trinomial. $$64 x^{2}-16 x+1$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(64 x^{2}-16 x+1\) is \((8x - 1)^{2}\).

Step 1: Identify the Terms

The first step is to identify the terms \(a\), \(b\) and \(c\) from the given expression \(64 x^{2}-16 x+1\). Here, \(a^{2}\) is \(64x^{2}\), which implies \(a\) is \(8x\). Also, \(b^{2}\) is \(1\), so \(b\) is \(1\). And \(2ab\) is \(-16x\).

Step 2: Confirm If It's a Perfect Square Trinomial

One identifying factor for a perfect square trinomial is that \(2ab\) equals the second term. So, in this case, \(2ab\) equates to \(2*8x*1=-16x\). Since this is correct, we can confirm that this is indeed a perfect square trinomial.

Step 3: Factor the Trinomial

Following the perfect square trinomial formula \((a + b)^{2}\), and using the values of a and b obtained from Step-1 (\(a = 8x\) and \(b = 1\)), the factored form is \((8x - 1)^{2}\).