Problem 28 - College Algebra

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}+16 x $$

Step-by-Step Solution

Answer

The constant that must be added to \(x^{2}+16x\) to form a perfect square trinomial is 64. The trinomial becomes \(x^{2}+16x+64\) and this factorizes to \((x+8)^{2}\)

Step 1: Find the Missing Constant in a Perfect Square Trinomial

For a trinomial to be a perfect square trinomial, the constant term has to be the square of half the coefficient of the 'x' term. Here, the coefficient of 'x' is 16. So, half of it is 8. Thus, the constant you have to add will be \(8^{2}\) which equals 64.

Step 2: Write down the new trinomial

Adding the calculated constant to the given binomial, the perfect square trinomial becomes \(x^{2}+16x+64\)

Step 3: Factorize the perfect square trinomial

This trinomial is a perfect square so it can be factored as the square of a binomial. To factorize it, you need to find two numbers whose sum gives you the coefficient of the 'x' term (which is 16 in this case), and whose product provides the constant term (which is 64 in this case). The two numbers fulfilling these conditions are 8 and 8. So, \(x^{2}+16x+64\) factorizes to \( (x+8)^{2}\)

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This process is particularly helpful in algebra when solving quadratic equations and helps to simplify expressions. The main idea is to make the quadratic expression into a format that can be easily factored.

Let's walk through an example: we start with the expression \(x^2 + 16x\). To complete the square, we look for the constant that makes this expression a perfect square trinomial. The key step here is understanding how to calculate this constant:

• Take the coefficient of the \(x\) term, which in our case is 16.
• Divide it by 2; here, we get 8.
• Square the result. That means you calculate \(8^2 = 64\).

By adding 64 to the binomial \(x^2 + 16x\), we can transform it into the trinomial \(x^2 + 16x + 64\), which is a perfect square.

Factoring Trinomials

Factoring trinomials, especially a perfect square trinomial, is usually simpler than it looks. A perfect square trinomial is one where the binomial squared gives you this trinomial. Recognizing and factoring such trinomials is a crucial skill in algebra.

In our example, the trinomial \(x^2 + 16x + 64\) can be factored easily:

• You need two identical numbers that add to give the middle term's coefficient (16) and multiply to give the last term (64).
• For \(x^2 + 16x + 64\), the numbers 8 and 8 meet these criteria: \(8 + 8 = 16\) and \(8 imes 8 = 64\).

This allows us to factor the trinomial into \((x + 8)^2\), which is much simpler to work with, especially when solving equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Understanding and manipulating these expressions is key to mastering algebra and other higher math topics.

In this example, the expression \(x^2 + 16x\) is an algebraic expression that isn't a complete polynomial yet; it's sometimes referred to as a binomial because it has two terms. Adding a constant to turn it into a trinomial—and then into a perfect square trinomial—is a classic problem in algebra.

Working with algebraic expressions often involves rearranging and simplifying them. When dealing with quadratic expressions like ours, it typically involves completing the square or factoring to simplify further or solve equations. Thus, learning how to maneuver these expressions by methods like completing the square plays a pivotal role in progressing through algebraic problems efficiently.