Problem 6 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}+3 x\)

Step-by-Step Solution

Answer

The equation after completing the square and then factoring results in \( \left( x+\frac{3}{2} \right)^{2} - \frac{9}{4}\).

Step 1: Completing the Square

First we need to determine the value that will complete the square. To do this, take half of the coefficient of x, and square it. In this case, the coefficient is 3, so half of it is \( \frac{3}{2} \), and its square is \( \left( \frac{3}{2} \right)^{2} = \frac{9}{4}\). Add and subtract this value to the equation. Now the equation becomes, \(x^{2}+3x+\left(\frac{3}{2}\right)^{2} - \left(\frac{3}{2}\right)^{2}\), which simplifies to, \(x^{2}+3x+\frac{9}{4}-\frac{9}{4}\).

Step 2: Form Perfect Square Trinomial

Grouping the first three terms and leaving the subtracted squared value outside, we get, \( \left( x^{2}+3x+\frac{9}{4} \right) - \frac{9}{4}\). This simplifies to \( \left( x+\frac{3}{2} \right)^{2} - \frac{9}{4}\), which is a perfect square trinomial.

Step 3: Factoring the Trinomial

The perfect square trinomial can't be factored as it already is in its simplest form; thus, the equation remains \( \left( x+\frac{3}{2} \right)^{2} - \frac{9}{4}\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special kind of algebraic expression. It looks like a squared binomial and is always in the form of \( (a + b)^2 = a^2 + 2ab + b^2 \). When you complete the square, you're rewriting a quadratic expression so that it becomes a perfect square.

• The importance of a perfect square trinomial is that it makes solving quadratic equations easier, especially with the help of the quadratic formula or during factorization.
• To achieve a perfect square trinomial, you have to add a specific value that completes the square in the quadratic expression.
• For example, in the exercise \( x^2 + 3x \), we determined that \( \left( \frac{3}{2} \right)^2 = \frac{9}{4} \), needs to be added to make it a perfect square trinomial.

Understanding how to create and recognize perfect square trinomials is crucial in mastering the process of solving quadratic equations by completing the square.

Binomial

A binomial is a polynomial with exactly two terms. In the exercise, we started with a binomial, \( x^2 + 3x \). This simple form can be restructured into a trinomial by using methods like completing the square.

• Each term in a binomial can have a different degree or power. For example, \( x^2 + 3x \) has terms of degree 2 and degree 1.
• Transforming a binomial into something we can more easily handle, like a perfect square trinomial, is very helpful in algebra.
• Being aware of the structure of the binomial and the relationships between its terms help in rewriting and simplifying expressions.

Identifying and manipulating binomials forms the bedrock of algebraic processes like factoring and completing the square.

Factoring

Factoring is a key algebraic method used to simplify expressions, solve equations, and understand mathematical properties of numbers and expressions.

• When we talk about factoring in the context of completing the square, we're referring to expressing the completed perfect square trinomial in the form \((x + a)^2\) or similar.
• After completing the square in the exercise, the resulting perfect square trinomial \((x + \frac{3}{2})^2\) already represents the factorized form.
• The idea behind factoring is to break down expressions or equations into products of simpler factors, making them easier to solve or evaluate.

Factoring is a foundational skill in algebra, often culminating in finding roots of equations or simplifying complex expressions.