Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The factored form of the trinomial is \((x - 7)^2\).

Step 1: Identify the form of the trinomial

The given trinomial is \(x^{2}-14x+49\). It can be noticed that this trinomial is in the form of \(a^2 - 2ab + b^2\).

Step 2: Identify the values of a and b

Compare the trinomial with the form \(a^2 - 2ab + b^2\) to identify the values of \(a\) and \(b\). Here, \(a = x\) (since \(a^2 = x^2\)), and \(b = 7\) (since \(b^2 = 49\)). Our task is to check whether \(2ab = -14x\) or not. With \(a = x\) and \(b = 7\), we have \(2ab = 2*x*7 = 14x\), so indeed our original trinomial is in the form of a perfect square trinomial.

Step 3: Apply the factoring formula

When the trinomial is in the form \(a^2 - 2ab + b^2\), it can be factored into \((a - b)^2\). Substituting the values of \(a\) and \(b\) in this formula, we get \((x - 7)^2\).