Problem 51 - College Algebra Essentials

Question

Factor each perfect square trinomial. $$x^{2}-14 x+49$$

Step-by-Step Solution

Answer

The factorized form of the perfect square trinomial \(x^{2}-14x+49\) is \((x-7)^{2}\)

Step 1: Identify the trinomial

Firstly, identify that the provided expression \(x^{2}-14x+49\) is indeed a trinomial, because it has three terms.

Step 2: Compare with standard perfect square trinomial form

The standard form of a perfect square trinomial is \(a^{2}-2ab+b^{2}\). Compare the given trinomial \(x^{2}-14x+49\) with this standard form. Here, \(a=x\), \(2ab=14x\) (which implies \(b=7\)), and \(b^{2}=49\). Therefore, the given trinomial can be expressed in the standard form.

Step 3: Factor the trinomial

Given that this is a perfect square trinomial, it can be factored into \((a-b)^{2}\). Here, replacing \(a\) with \(x\) and \(b\) with \(7\), let's factorize \(x^{2}-14x+49\) into its factors: \((x-7)^{2}\)