Problem 37 - College Algebra Essentials

Question

In Exercises \(35-46,\) determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-10 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial $(x^{2} - 10x)$ to form a perfect square trinomial is \(25\), and the trinomial can further be factored to \((x - 5)^2\).

Step 1: Identify the Coefficient

First, it is important to identify the coefficient of \(x\). In this case, the coefficient of \(x\) is \(-10\). We have \(a = 1\) and, \(-2ab = -10x\). From this we can find the value of \(b\) which is \(-10 / -2 = 5\). Therefore, \(b = 5\).

Step 2: Calculate the Constant

Next, we have to calculate \((b)^2\) to find the constant that should be added. Squaring \(5\) provides the constant needed to make the binomial a perfect square trinomial. Doing so gives us \((5)^2 = 25\). So, the constant we need to add to make it a perfect square trinomial is \(25\).

Step 3: Construct the Perfect Square Trinomial and Its Factored Form

Adding the constant to the binomial to get our perfect square trinomial as \(x^{2} -10x + 25\). To factor the perfect square trinomial, check if it takes the form \((a - b)^2\). The number \(25\) is the square of \(5\) as computed earlier so this checks out. The factorization becomes \((x - 5)^2\).

Key Concepts

Understanding Binomials

A **binomial** is a type of algebraic expression. It contains two terms. These terms can include numbers, variables, or both, separated by a plus "+" or minus "-" sign.
In the exercise given, the binomial is expressed as \( x^2 - 10x \). The key elements are:

• \( x^2 \) is the first term.
• \( -10x \) is the second term.

This algebraic expression is called a binomial because there are exactly two terms present.

Introduction to Factoring

**Factoring** is a critical process in algebra that simplifies expressions for easier manipulation and solution. It involves breaking down complex expressions into simpler factors or parts.
When we factor an expression, we write it as a product of its factors. In the exercise, we converted the expression \( x^2 - 10x + 25 \) into its factored form \((x - 5)^2\).
Factors are like the building blocks of an equation. Knowing how to factor efficiently allows you to solve and visualize algebraic problems better.

Exploring Algebraic Expressions

**Algebraic expressions** combine numbers, variables, and operators into meaningful mathematical statements. They form the foundation of algebra and can represent quantities in real-world scenarios.
An algebraic expression could be as simple as a single number or variable or as complex as a polynomial with multiple terms. Our given expression \( x^2 - 10x \) is an example of an expression formed by variables and coefficients.

• Constants like \( -10 \) give specific values.
• Variables, represented by \( x \), stand for unknowns.

Understanding how to manipulate these expressions is key in algebra.

Completing the Square

**Completing the square** is a method used to transform a quadratic trinomial into a perfect square trinomial. This is a valuable technique in algebra for solving equations, particularly quadratics.
In our exercise, the binomial \( x^2 - 10x \) becomes a perfect square trinomial \( x^2 - 10x + 25 \) by adding the constant \( 25 \). The term \( 25 \) makes the expression into a square of \((x - 5)^2\).
To complete the square means to find the right constant term to add to make the expression a square and then rewrite it in its squared form, where the expression is easier to solve or analyze.