Problem 42 - 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. $$ x^{2}-9 x $$

Step-by-Step Solution

Answer

The constant to be added to the binomial to make it a perfect square trinomial is 20.25, and the factored form of the trinomial is \(( x - 4.5 ) ^ { 2 }\).

Step 1: Identify a and b

In the formula for a perfect square trinomial \((a-b)^2 = a^2 - 2ab + b^2\), here, in the binomial \(x^{2}-9 x\), \(a= x\) and \( -2ab= -9x\). Hence, \(b= \frac{-9x}{-2x}\) which gives \(b=4.5\).

Step 2: Find the value for \(b^2\)

\Now, compute the square of \(b\), i.e., \(b^2 = (4.5)^2 = 20.25 \). This is the constant that should be added to the binomial \(x^{2}-9 x\) to make it a perfect square trinomial.

Step 3: Write and factor the trinomial

Add the determined value, \(20.25\), to the given binomial, we get a trinomial, which is \(x^{2}-9x + 20.25 \). Now, factor this trinomial, it factors as \(( x - 4.5 ) ^ { 2 }\) as it fits the perfect square trinomial formula \((a-b)^2\).

Key Concepts

Factoring Trinomials

Understanding how to factor trinomials is a foundational skill in algebra. It involves rewriting a trinomial, which is a polynomial with three terms, into a product of two binomials. The process becomes particularly straightforward when dealing with a perfect square trinomial because it follows a specific pattern.

Consider the general form of a perfect square trinomial \(a^2 \pm 2ab + b^2\), which factors into \( (a \pm b)^2 \). When tackling a problem like \(x^2 - 9x\), the goal is to add a constant to create such a trinomial. To find the correct constant, we seek a value \(b\) that, when squared, completes the trinomial and allows us to write it as a squared binomial. Following the pattern, \(x^2 - 9x + 20.25\) factors into \( (x - 4.5)^2\), demonstrating a perfect application of factoring trinomials in its simplest form.

Using this approach not only helps in factoring exercises but also in solving quadratic equations and grasping the fundamentals of the algebraic structure of polynomials.

Completing the Square

The technique of completing the square makes it possible to transform any quadratic equation into a perfect square trinomial, which in turn is easily solvable. To complete the square, one must find a value that, when added to the expression, allows it to be written as the square of a binomial.

In our example, we start with \(x^2 - 9x\). To turn this into a perfect square, we look at the coefficient of \(x\), which is \(9\), and then use the relation \(2ab\) to find \(b\) by dividing the coefficient by \(2\). Afterwards, we square \(b\) to get the necessary constant to add. Specifically, for \(x^2 - 9x\), the value \(4.5^2\) or \(20.25\) completes the square, so the expression \(x^2 - 9x + 20.25\) is now a perfect square trinomial that factors into \( (x - 4.5)^2\).

This method is particularly important for solving quadratics and also plays a critical role in the development of the quadratic formula, hence deepening the understanding of algebraic manipulation and equations.

Algebraic Expressions

An algebraic expression is a mathematical phrase representing a combination of numbers, variables (typically letters), and arithmetic operations like addition, subtraction, multiplication, and division. In cases like \(x^2 - 9x\), you're working with a binomial expression, which contains two terms. Trinomials are an extension of this concept, with three terms.

Algebraic expressions are the building blocks for formulating and solving equations. Understanding how to manipulate these expressions by adding, subtracting, multiplying, dividing, or factoring is key to mastering algebra. The beauty of algebra comes from how these expressions can model real-world problems, allowing for variables to represent unknown quantities and providing a systematic way of finding these unknowns.

Perfect square trinomials, and other special products, are excellent examples of how patterns in algebraic expressions can simplify complex problems and establish a foundation for exploring more advanced mathematical concepts.