Problem 37 - Algebra and Trigonometry

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. $$ x^{2}-10 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial \(x^{2}-10x\) to make it a perfect square trinomial is 25. The resulting perfect square trinomial is \(x^{2}-10x+25\), which factors to \((x-5)^2\).

Step 1: Identify the coefficients

In the given binomial, \(x^{2}-10x\), the coefficient of x^2 is 1 (a^2) and the coefficient of x is -10 (2ab).

Step 2: Calculate the coefficient 'b'

Here a is 'x' and 2ab is '-10x'. We interpret this as 2*b*x = '-10x'. To find 'b', we equate the coefficients of x which gives '2b = -10', therefore, b=-5.

Step 3: Use the value of 'b' to complete the square

The constant that should be added to the binomial is \(b^2\). Substitute b=-5 into \(b^2\) which gives \(25\).

Step 4: Write and factor the perfect square trinomial

The perfect square trinomial obtained by adding the constant is \(x^2 - 10x + 25\). The factored form of this trinomial is \((x - 5)^2\).

Key Concepts

Understanding Binomials

A binomial is a type of polynomial that consists of exactly two terms. It can take various forms due to the inclusion of variables and coefficients. In algebra, a common task is to manipulate binomials in a way that makes them more useful for solving equations or modeling data.

Let's consider the example given, \(x^2 - 10x\). Here, we have two terms: \(x^2\) and \(-10x\). Notice that this binomial contains a variable term, \(x^2\), and a linear term, \(-10x\).

These components are key to many algebraic operations such as factoring, which we will discuss next. Recognizing the structure of a binomial helps predict how it can be transformed and simplified.

Factoring Overview

Factoring a polynomial is the process of breaking it down into simpler "factors" that, when multiplied together, give the original polynomial. This technique is essential, especially when dealing with quadratic expressions, like our perfect square trinomial.

In our exercise, the goal is to start with the binomial \(x^2 - 10x\) and transform it into a perfect square trinomial by factoring. Once we determine the necessary constant to add, we create \(x^2 - 10x + 25\).

The next step is to factor this trinomial into the square of a binomial, yielding \((x - 5)^2\).

Through factoring, complex equations can be simplified, making them much easier to solve or analyze. This process often involves patterns and techniques like completing the square, which we will explore further.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial, making it easier to solve or factor. This method involves creating a trinomial that is the square of a binomial.

For the binomial \(x^2 - 10x\), we first need to identify the middle coefficient, \(-10\), and its relationship to the binomial. By identifying \(b\) using the equation \(2b = -10\), we solve for \(b = -5\).

To "complete the square," we take this \(b\) and find \(b^2\), resulting in \(25\). Adding this constant transforms our binomial into a perfect square trinomial: \(x^2 - 10x + 25\). This trinomial is then easily factored into \((x - 5)^2\).

Mastering completing the square allows you to rewrite equations in a way that reveals solutions more clearly.

Coefficients in Quadratics

Coefficients are numerical or constant factors before variables in an algebraic expression. Understanding how to identify and manipulate coefficients is critical in algebra.

In the example \(x^2 - 10x\), the coefficient of \(x^2\) is \(1\) (implicit), and the coefficient of \(x\) is \(-10\). These coefficients guide the process of completing the square.

By focusing on \(-10\), we determine how to adjust the binomial into a perfect square trinomial. The relationship \(2b = -10\) helps find \(b\). Once we determine \(b\), coefficients allow us to calculate the necessary adjustment, \(b^2 = 25\), transforming the binomial into the trinomial for easy factoring.

Recognizing these coefficients and their roles gives you control over manipulating quadratic equations efficiently.