Problem 52 - Algebra and Trigonometry

Question

Factor each perfect square trinomial. $$x^{2}-10 x+25$$

Step-by-Step Solution

Answer

\((x - 5)^{2}\

Step 1: Identify a and b values

In our trinomial, \(a\) would be the square root of the first term, \(x^{2}\) which is \(x\), and \(b\) would be the square root of the third term, 25, which is 5.

Step 2: Verify the fit to the perfect square trinomial form

Perfect square trinomials fit the form \(a^{2}\)-2ab+\(b^{2}\). The term in the middle, -10x, should equal to -2ab = -2*\(x\)*5 = -10x, and it does. So, this verifies that the original expression we have is indeed a perfect square trinomial.

Step 3: Use the Square of a Binomial rule

Using the square of a binomial rule, which states that \((a-b)^{2} = a^{2} - 2ab + b^{2}\), we can rewrite our trinomial. Considering our identified \(a\) and \(b\) values, our original equation \(x^{2} - 10x + 25\) is equal to \((x - 5)^{2}\).