Problem 41 - Introductory Algebra for College Students

Question

Explain how to complete the square for a binomial. Use \(x^{2}+6 x\) to illustrate your explanation.

Step-by-Step Solution

Answer

The completed square form of the given expression \(x^{2} + 6x\) is \((x+3)^{2}\).

Step 1: Identify the quadratic expression.

For the given quadratic expression \(x^{2} + 6x\), the coefficients of \(x^{2}\) and \(x\) are 1 and 6 respectively.

Step 2: Determine the term needed to complete the square.

To complete the square, the formula \((b/2)^{2}\) is employed, where 'b' is the coefficient of 'x'. In this case, \(b=6\). Substituting the value of 'b' into the formula gives \((6/2)^{2}=(3)^{2}=9\).

Step 3: Form a perfect square trinomial

Add the determined value to the given quadratic expression to form a perfect square trinomial. Here, we add 9 to \(x^{2} + 6x\) to get \(x^{2} + 6x + 9\).

Step 4: Factor the perfect square trinomial

Factor the perfect square trinomial, \(x^{2} + 6x + 9\), which becomes \((x+3)^{2}\). That value is the completed square form of the original expression.