Problem 7 - Introductory Algebra for College Students

Question

Complete the square for binomial. Then factor the resulting perfect square trinomial. \(x^{2}-7 x\)

Step-by-Step Solution

Answer

The binomial \( x^2 - 7x \), after completing the square and factoring the resulting perfect square trinomial, is \( (x - 3.5)^2 \).

Step 1: Complete the Square

A perfect square trinomial can be created from a given binomial by adding the square of half the coefficient of the linear term. That is, \( (b/2)^2 \) where \( b \) is the coefficient of \( x \). Our given binomial is \( x^2 - 7x \). So \( b = -7 \). The square of half of \( b \) will be \( (-7/2)^2 = 12.25 \). Hence, to complete the square for \( x^2 - 7x \), add 12.25 to it. So, our perfect square trinomial is \( x^2 - 7x + 12.25 \).

Step 2: Factor the Perfect Square Trinomial

The resulting perfect square trinomial, \( x^2 - 7x + 12.25 \), can be factored by expressing it as square of a binomial, which is in the form \( (x – c)^2 \), where \( c = b/2 \) and \( b = -7 \). Hence, \( c = -7/2 = -3.5 \). So, the factored form of the perfect square trinomial is \( (x - 3.5)^2 \).

Key Concepts

Perfect Square Trinomial

Understanding the notion of a perfect square trinomial is fundamental in algebra. It involves recognizing a trinomial that can be expressed as the square of a binomial. This occurs when the trinomial takes the form of \( a^2 + 2ab + b^2 \), where \( a \) and \( b \) are real numbers or algebraic expressions. To visualize, imagine folding a binomial square perfectly along its diagonal to get a duplicate of two rectangles and a smaller square, giving you the components of a perfect square trinomial.

When students complete the square, they are essentially creating a perfect square trinomial from a given binomial. By adding a specific value, which is the square of half the linear coefficient, the expression can be neatly factored into a binomial square, \( (x+c)^2 \), streamlining further calculation or graphing tasks related to quadratic functions.

Factoring Binomials

Factoring is akin to breaking down numbers into their multiplicative building blocks, and with algebraic expressions like binomials, it's about finding an equivalent expression that represents the product of two or more factors. One of the most satisfying results occurs when factoring leads to a perfect square binomial.

Imagine you're tasked with unpacking a neatly wrapped gift box. You're taking apart the adhesive tape, which holds the flaps together, to reveal the contents inside. Similarly, understanding how to factor binomials into their constituent parts allows you to delve deeper into the equation's characteristics. Factoring a perfect square trinomial back into a binomial is like reversing the wrapping process, providing you a compact and operable mathematical 'parcel', like \( (x-3.5)^2 \).

Algebraic Expressions

An algebraic expression is a collection of numbers, variables, and operations combined without an equality sign. Unlike equations, these expressions aren't about finding a specific value but understanding relationships and potential transformations. They form the core language of algebra and can become as simple or complex as the situation demands.

Students should visualize algebraic expressions as sentences where the numbers and variables are the 'words', and the operations are the 'grammar' rules that define how the words can be combined. It's through the manipulation of these expressions, such as completing the square or factoring, that one can simplify complexities, solve equations, or describe geometric figures in algebraic terms.