Problem 28 - Algebra 1: Concepts and Skills

Question

PERFECT SQUARES Factor the expression. $$x^{2}-20 x+100$$

Step-by-Step Solution

Answer

The expression \( x^{2}-20x+100 \) is a perfect square and can be factorized to \( (x-10)^{2} \)

Step 1: Identify the coefficients and compare with the perfect square trinomial

Firstly, it's important to identify the coefficients in the given expression. Notice that the expression \( x^{2}-20x+100 \) looks similar to the form of \( (x-a)^{2} \), which simplifies to \( x^{2}-2ax+a^{2} \). Thus, comparing the coefficients of the given expression with the standard perfect square trinomial, it can be deduced that \( a^{2} = 100 \) and \( -2a = -20 \). Solve for a to get \( a = 10 \).

Step 2: Factorize the expression

The value of a obtained in step 1 will be used to factorize the expression. Substituting a in the form \( (x-a)^{2} \), the factored form of the expression becomes \( (x-10)^{2} \)

Key Concepts

Understanding Perfect Square Trinomials

A perfect square trinomial is a special type of quadratic expression that can be factored into a squared binomial. It has the general form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), where \(a\) and \(b\) are real numbers. This form is derived from the square of a binomial, \( (a + b)^2 \), which gives \(a^2 + 2ab + b^2\), or \( (a - b)^2\), which gives \(a^2 - 2ab + b^2\).

When factoring a perfect square trinomial, look for two identical terms squared and a middle term that is precisely twice the product of the terms being squared. If these conditions are met, the expression can be neatly written as the square of a binomial. This simplifies calculations and makes understanding the structure of algebraic expressions clearer.

Breaking Down Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In an expression like \(x^{2}-20x+100\), \(x\) is the variable, the numbers are the coefficients, and the operations involved are addition and subtraction. Intuitively, one could see algebraic expressions as recipes that tell you how to mix these ingredients to create algebraic statements.

Understanding the structure of these expressions is key to mastering algebra. By identifying patterns within them, such as perfect square trinomials, and applying the proper factoring techniques, you unlock a streamlined way to simplify complex problems and find solutions to equations.

The Essentials of Factoring Quadratics

Factoring quadratics, a central concept in algebra, involves rewriting a quadratic expression as the product of two binomials. The standard form of a quadratic is \(ax^{2} + bx + c\), but not all quadratics are created equal. Some have special forms that allow for shortcuts in factoring.

One such shortcut occurs with perfect square trinomials, where the quadratic can be factored into the square of a binomial. Recognizing perfect squares quickly can save time and effort, making problem-solving more efficient. Factoring quadratics is not just a mechanical process but also a stepping stone to further mathematical exploration, like solving quadratic equations.

Coefficient Comparison Method

Coefficient comparison is a technique used to identify and match the coefficients of equivalent algebraic expressions. It requires a keen eye to notice patterns and relationships between different parts of an equation. For instance, in a perfect square trinomial such as \(x^{2} - 20x + 100\), the coefficient of \(x^{2}\) is implied to be 1, the coefficient of \(x\) is -20, and the constant term is 100.

By comparing these coefficients to the standard form of a perfect square \( (x-a)^{2} \), we deduce that \( -2a\) should equal -20 and \(a^{2}\) should be 100. Solving for \(a\) confirms that \(a = 10\). This methodical approach is fundamental when factoring perfect square trinomials and is a valuable tool for solving more intricate algebraic problems.