Problem 42 - 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 to be added is (81/4) that transforms the given binomial into the perfect square trinomial \(x^{2}-9x+(81/4)\). The factored form is \((x-9/2)^{2}\).

Step 1: Identify \(a\) and \(b\)

In the given binomial, \(x^{2}-9x\), \(a\) have been identified as \(x\) and \(2ab\) as \(9x\). This means \(2ab = 9x\), or \(2b = 9\) after dividing both sides by \(a\) which is \(x\). Therefore, \(b\) is equal to \(9/2\).

Step 2: Calculate \(b^{2}\)

After identifying the value of \(b\) which is \(9/2\), calculate \(b^{2}\) because the desired perfect square trinomial will be in the form \(x^{2}-9x+b^{2}\). Hence, \(b^{2} = (9/2)^{2} = 81/4\).

Step 3: Add \(b^{2}\) to the binomial

Having determined that \(b^{2}\) equals 81/4, add this constant to the binomial to form a perfect square trinomial. The result will be \(x^{2}-9x+(81/4)\).

Step 4: Factorize the trinomial

Finally, the perfect square trinomial can be factored to the form \((a-b)^{2}\). In this case, factor the trinomial to obtain \((x-9/2)^{2}\). This is the final factored form.

Key Concepts

Understanding Binomials

In algebra, a binomial is an algebraic expression that consists of exactly two terms.These terms are usually joined by a plus or minus sign.For example, in the expression given in the exercise, \(x^2 - 9x\), there are two terms: \(x^2\) and \(-9x\).

To make binomials more manageable, especially for further operations, it's essential to recognize them and understand their parts.In this example:

• The first term is a square, \(x^2\)
• The second term includes the variable \(x\), indicating a linear term, \(-9x\)

Identifying these will help you manipulate them into different forms, like turning them into a trinomial as we will see next. Grasping these basics is fundamental when working with polynomial expressions.

Factoring Trinomials

Factoring trinomials is a crucial skill in algebra. It involves rewriting a trinomial as a product of two binomials.For a trinomial to be a perfect square, it often appears in the form \(a^2 - 2ab + b^2\), or \((a-b)^2\), once factored.

In the example, after finding the necessary constant to add (\(\frac{81}{4}\)), the trinomial \(x^2 - 9x + \frac{81}{4}\) is formed.The goal is to express this as a squared binomial.

Steps to factor:

• Identify the resulting components: the first square, double the product, and the last square.
• Write down the squared binomial: in this exercise, it is \((x - \frac{9}{2})^2\).

This process simplifies expressions and is useful for solving equations involving trinomials.

Exploring Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations.They can range from simple expressions like \(x + 1\) to more complex combinations involving multiple variables and powers.

In this exercise, the focus was on transforming an algebraic expression from a binomial into a perfect square trinomial.Understanding how these expressions work, especially their properties and how to manipulate them, is essential in algebra.

Key aspects include:

• Recognizing different types of expressions (monomial, binomial, trinomial)
• Knowing their individual components: coefficients (numbers) and terms (variables and powers)
• Being able to perform operations like addition, subtraction, and factorization on them

Mastering the use of algebraic expressions allows for solving more complex problems, understanding sequences, and even tackling equations in higher math.