Problem 34 - College Algebra

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-9 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial \(x^2 - 9x\) to make it a perfect square trinomial is 20.25. The trinomial is \(x^2 - 9x + 20.25\) and can be factored as \((x - 4.5)^2\).

Step 1: Identifying the Coefficients

Identify the coefficient of the first term (\(a^2\)) and the second term (\(-2ab\)). In the binomial \(x^{2}-9x\), \(a=x\) and the coefficient of \(x\) in the term \(-9x\) is \(-9\).

Step 2: Calculate the value of b

We need to find \(b\) from the coefficient of \(x\), which is \(-9\). Since the coefficient is of the form \(-2ab\), we equate \(-2ab\) to \(-9\). In this case, \(a=x\) and \(b?\) is what we're solving for. This gives \(-2xb = -9\), which simplifies to \(b = -9/(-2x) = 4.5/x\).

Step 3: Determine the constant term

In the perfect square trinomial, the constant term is \(b^2\). To find \(b^2\), square the \(b\) value we found in the last step. \(b\) = \(4.5/x\), therefore, \(b^2 = (4.5/x)^2 = 20.25/x^2\). As we're looking for a constant to make the expression a perfect square, our result should be agnostic of \(x\). As a result, when we multiply by \(x^2\) to remove the denominator, we got \(b^2 = 20.25\). This is the constant that should be added to the binomial to make it a perfect square trinomial.

Step 4: Write and Factor the Trinomial

Write out the new trinomial, which would be \(x^2 - 9x + 20.25\). This is a perfect trinomial square and can be factored as \((x - 4.5)^2\).

Key Concepts

Perfect Square Trinomial

A **perfect square trinomial** is an algebraic expression that results from squaring a binomial. Think of it as a neat package formed by multiplying a binomial by itself. Perfect square trinomials often come in the form of \(a^2 - 2ab + b^2\) or \(a^2 + 2ab + b^2\). These trinomials are special because they can be expressed as the square of a single binomial.

In the given exercise, we turned the binomial \(x^2 - 9x\) into the perfect square trinomial \(x^2 - 9x + 20.25\). To achieve this, we identified the constant needed to complete the square, ensuring the expression could be written as \((x - 4.5)^2\).

• Here, the initial binomial lacked a constant term, but by calculating \(b^2\) as 20.25, we completed the trinomial into a perfect square.
• We then rewrote the completed trinomial as a squared binomial, \((x - 4.5)^2\), making it easy to factor.

Recognizing and forming perfect square trinomials is a handy skill for simplifying and solving quadratic equations.

Factoring

**Factoring** is the process of breaking down an expression into simpler components or products. This method is crucial in solving quadratic equations and simplifying algebraic expressions.

When factoring perfect square trinomials, the goal is to rewrite them as the square of a binomial. Once you've confirmed the trinomial matches the pattern \(a^2 \pm 2ab + b^2\), rewriting it involves identifying the values of \(a\) and \(b\) and expressing the trinomial as \((a \pm b)^2\).

• In our case, after completing the square, the trinomial \(x^2 - 9x + 20.25\) factored into \((x - 4.5)^2\).
• This makes solving and simplifying so much easier, as solving squared binomials takes much less effort than dealing with full trinomials.

Mastering factoring, especially for perfect squares, significantly aids problem-solving in algebra. It simplifies complex expressions and paves the way for understanding polynomials and equations.

Algebraic Expressions

An **algebraic expression** consists of constants, variables, and operations (like addition and multiplication) combined to form a meaningful mathematical phrase. These expressions are the building blocks of algebra and are used to represent numbers and solutions compactly.

In algebra, working with expressions like \(x^2 - 9x\) involves simplifying, factoring, and expanding forms to solve equations or evaluate mathematical models. Expressions allow us to transform problems from verbal statements into solvable mathematical forms.

• Completing the square, as seen in our exercise, is an integral part of manipulating algebraic expressions. It transforms the structure to reveal the inherent simplicity, giving us a solvable equation like \((x - 4.5)^2\).
• By recognizing and rewriting terms, you gain a deeper understanding of how numbers and terms interrelate in the context of algebra.

Understanding and manipulating algebraic expressions is essential for any further study in mathematics and science. It's the language of math, clarifying complex ideas and enabling problem-solving across multiple disciplines.