Problem 4 - Algebra 1

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}+50 x$$

Step-by-Step Solution

Answer

The term that should be added is 625.

Step 1: Identifying the values of 'a' and 'b'

In the given expression, 'a' corresponds to the coefficient of \(x^2\), which is 1. 'b' corresponds to half of the coefficient of 'x', which is \(50/2 = 25\).

Step 2: Adding square of 'b' to the expression

To make it a perfect square trinomial, we add the square of 'b' to the expression, which gives us \(x^{2} + 50x + 25^2\).

Step 3: Completion

After adding the square of 'b' to the original expression, we now have a perfect square trinomial. Therefore, the term that should be added to the original expression is \(b^2 = 25^2 = 625\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They can be simple, like a single number or variable, or more complex, involving multiple terms. For example, the expression given in the exercise, \(x^2+50x\), includes variables and a coefficient. Here:

• \(x^2\) represents a term where \(x\) is squared.
• \(50x\) includes the variable \(x\) with a coefficient 50.

Understanding each part of an algebraic expression is crucial, especially when performing operations like addition or factorization. In this case, we are primarily concerned with transforming the expression into a more convenient form known as a "perfect square trinomial." This involves modifying the existing terms or adding new terms to achieve a specific mathematical structure.

Polynomials

Polynomials are a type of algebraic expression consisting of variables raised to various powers and multiplied by coefficients. A general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). Each term in this format is made up of a coefficient, variable, and exponent. For example, \(x^2 + 50x\) is a polynomial with two terms:

• \(x^2\) with a coefficient of 1 and an exponent of 2.
• \(50x\) with a coefficient of 50 and an exponent of 1.

When dealing with polynomials, one important concept is that of a perfect square trinomial, which can be expressed as \((ax+b)^2\). It simplifies to \(ax^2 + 2abx + b^2\). Recognizing this pattern is key to completing the square and solving quadratic equations.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This method is especially useful for solving quadratic equations or helping to rewrite expressions more elegantly. The given problem requires finding the right term to convert \(x^2 + 50x\) into a perfect square trinomial.

How to Complete the Square:

• Identify the term coefficient in front of \(x\). This is 50 in our case.
• Take half of this coefficient, which is \(\frac{50}{2} = 25\).
• Square this result, giving you \(25^2 = 625\).
• Add this squared term to the original expression to complete the square: \(x^2 + 50x + 625\).

By completing the square, the expression \(x^2 + 50x\) is now \((x + 25)^2\) after adding 625. This technique not only helps in simplifying expressions but also plays a pivotal role in solving equations and understanding the properties of parabolas in graphing.