Problem 52 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given values of the variables. (Lesson 1.2 ) \((b-c)^{2}\) when \(b=2\) and \(c=1\)

Step-by-Step Solution

Answer

The value of the expression \((b-c)^{2}\) for \(b=2\) and \(c=1\) is \(1\).

Step 1: Understand the expression

The given expression is \((b-c)^{2}\), which represents the squaring of the binomial '(b-c)'. This involves two operations - subtraction, 'b' minus 'c', and then squaring the result.

Step 2: Substitute the given values

Given that \(b=2\) and \(c=1\), the next step is to substitute these values into the binomial. This results in \((2-1)^{2}\).

Step 3: Simplify the binomial

Simplify the expression inside the parentheses first, following the order of operations (also known as BIDMAS/BODMAS/PEDMAS). This yields \(1^{2}\).

Step 4: Square the result

The final step is to square the result from Step 3. Squaring a number is the same as multiplying the number by itself. Thus, \(1^{2}\) results in \(1\).

Key Concepts

Substitution in Expressions

Substitution in expressions is like filling in a puzzle. When you have an expression with variables, it's like having placeholders where something else can fit in. You substitute these placeholders with actual numbers given in a problem.

Let's take an example from the exercise: in the expression \((b-c)^2\), "b" and "c" are variables. If you're given that \(b = 2\) and \(c = 1\), you replace every "b" with 2 and every "c" with 1. It's that simple! The expression then looks like this: \((2-1)^2\).

• Find the variables in the expression.
• Replace each variable with the number provided.
• You're ready to tackle the next steps in solving the expression!

Understanding substitution helps make algebra much easier. It allows you to transform abstract equations into something concrete that you can calculate.

Order of Operations

Order of operations is a set of rules that tells you the order to do mathematical calculations. You might have heard of PEMDAS, BODMAS, BIDMAS, or BEDMAS. These acronyms stand for Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction.

In our exercise, once you've substituted the numbers into \((2-1)^2\), the next step is to use the order of operations to solve it correctly. You start with what's inside the parentheses first, just like in step 3 of the solution. That means you perform \(2-1\) first, which gives you 1.

The steps typically are:

• Solve what's in parentheses/brackets first.
• Then do any exponents or orders (like squaring).
• Multiplication and division come next, from left to right.
• Finally, perform addition and subtraction, from left to right.

Getting the order right is crucial. It ensures everyone solves an expression in the same way and reaches the same answer.

Binomials

In algebra, a binomial is an expression with two terms joined by a plus or minus sign. The term "binomial" makes things sound complicated, but it just means two parts! For example, \((b-c)\) is a binomial because it consists of two terms, "b" and "c", separated by a minus.

In the exercise, our binomial \((b-c)\) becomes \((2-1)\) after substitution. Binomials are fundamental in algebra because they serve as the building blocks for more complex expressions and equations.

• Binomials involve two parts or terms.
• Each term is usually a number, variable, or both.
• They're connected by either a plus or minus sign.

Working with binomials requires using the correct order of operations and occasionally expanding them in problems. Understanding binomials is a great step towards mastering more advanced algebraic concepts!

Squaring Numbers

Squaring a number means you multiply it by itself. It's an exponent operation denoted by the small number 2 written as a superscript. For example, \(x^2\) means \(x\times x\).

In our exercise, you end up squaring "1" after calculating \((2-1)\). You need to find \(1^2\), which is simply \(1 \times 1\), resulting in 1. It sounds like magic, taking a number and turning it into something calculated!

Here's the essence of squaring:

• Take the number and multiply it by itself.
• Exponents make calculations simpler and shorten expressions.
• Useful in geometry, physics, and algebra to describe areas and other measurements.

Getting comfortable with exponents, especially squaring, is indispensable in math. They appear frequently in different aspects, and recognizing them makes your life a bit easier in solving problems.