Problem 81 - Introductory Algebra for College Students

Question

Factor completely. $$x^{2}-\frac{2}{5} x+\frac{1}{25}$$

Step-by-Step Solution

Answer

The factored form of the given expression is \((x-1/5)^2\).

Step 1: Identify the variables

In the given quadratic expression, identify the variables and constants - which are \(a=x\), \(b=\frac{1}{5}\) and \(c=a^2 - 2ab + b^2\). Here, \(a\), \(b\) and \(c\) are the coefficients of \(x^2\), \(x\) and the constant term respectively.

Step 2: Check for perfect square formula

Upon examining the given quadratic expression, one can see that it fits the formula of \(A^2 - 2AB + B^2\), where \(A = x\), \(B=a/b=1/5\) and the constant term \(c=a^2 - 2ab + b^2\). Hence it can also be written as: \((x-1/5)^2\)

Step 3: Factorization

So the quadratic expression factors into \((x-1/5)^2\).

Key Concepts

Perfect Square Trinomials

When it comes to factoring quadratics, recognizing a perfect square trinomial can be a game-changer. But what exactly is a perfect square trinomial? Imagine squaring a binomial, which means multiplying it by itself, like \( (x+y)^2 \). The result is a trinomial, which is a three-term polynomial, in the form of \( x^2 + 2xy + y^2 \). In a perfect square trinomial, this pattern is easy to identify because the first and last terms are perfect squares and the middle term is twice the product of the square roots of these perfect squares.

For instance, take the example \( x^2 - \frac{2}{5} x + \frac{1}{25} \). It closely resembles the pattern \( A^2 - 2AB + B^2 \), where \( A = x \) and \( B = \frac{1}{5} \). The first term, \( x^2 \), and the last term, \( \frac{1}{25} \), are both perfect squares. The middle term is exactly twice the product of \( x \) and \( \frac{1}{5} \), confirming that we have a perfect square trinomial. Therefore, this expression can be compactly written as \( (x - \frac{1}{5})^2 \), simplifying the factoring process.

Knowing how to identify and factor perfect square trinomials is a valuable skill because it frequently appears in algebra problems. Being able to quickly recognize this pattern allows for a smooth transition to solving quadratic equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols but does not have an equality sign like an equation. For example, the expression \( x^2 - \frac{2}{5} x + \frac{1}{25} \) is an algebraic expression. It's formed by combining the variables \( x \) with coefficients and constants in various ways, including addition, subtraction, multiplication, division, and exponents.

When you're working with such expressions, particularly quadratic expressions—which have the highest variable degree of 2—the goal is often to simplify or factor them. Factoring is like taking a complex expression and breaking it down into simpler, multiplied components. This process is helpful because it can make the expression easier to work with, whether you're solving an equation or graphing a function.

To effectively factor algebraic expressions, especially quadratics, you'll want to have a set of factoring techniques at your disposal. Recognizing patterns like the perfect square trinomial in the above exercise allows you to apply the appropriate technique quickly, thereby making the expression much simpler to handle.

Quadratic Equations

Moving on to the broader topic of quadratic equations, these are a step beyond algebraic expressions. A quadratic equation is a specific type of polynomial equation of the second degree, which means it has an \( x^2 \) term. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \( a \) cannot be zero. Solving a quadratic equation means finding the values of \( x \) that make this equation true, and there are several methods you can use to do so, such as factoring, completing the square, or using the quadratic formula.

Factoring is often the preferred first approach, as it can be the quickest and simplest method when applicable. For the given expression \( x^2 - \frac{2}{5} x + \frac{1}{25} \), if it were set equal to zero, the factored form \( (x - \frac{1}{5})^2 \) would reveal that the only solution to this particular quadratic equation is \( x = \frac{1}{5} \). Identifying that the equation involves a perfect square trinomial helps you to conclude with the solution without needing complex methods.

The ability to manipulate and solve quadratic equations is foundational for many areas of mathematics, and as such, understanding the various ways to factor quadratics, including recognizing perfect square trinomials, is an essential tool for any student’s mathematical toolkit.