Problem 49 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(x^{2}-x+1\) for a. \(x=2\) b. \(x=-3\)

Step-by-Step Solution

Answer

a. The expression evaluates to 3 for \(x=2\). b. The expression evaluates to 13 for \(x=-3\).

Step 1: Substitute Value for Part (a)

To evaluate the expression for \(x=2\), substitute \(2\) in the place of \(x\) into the expression \(x^2 - x + 1\). This gives us: \[ (2)^2 - 2 + 1 \]

Step 2: Calculate Each Term for Part (a)

First calculate \( (2)^2 \), which is \(4\). Then calculate \(-2\), which remains \(-2\). Finally add \(+1\). So, it becomes: \[ 4 - 2 + 1 \]

Step 3: Simplify the Expression for Part (a)

Simplify the expression by following the order of operations:\(4 - 2 = 2\).Now, \(2 + 1 = 3\).So, when \(x = 2\), the expression evaluates to 3.

Step 4: Substitute Value for Part (b)

For the second part with \(x = -3\), substitute \(-3\) in the place of \(x\) into the expression \(x^2 - x + 1\). This gives us: \[ (-3)^2 - (-3) + 1 \]

Step 5: Calculate Each Term for Part (b)

Calculate \((-3)^2\), which is \(9\). The next term, \(-(-3)\), simplifies to \(+3\). Lastly add \(+1\). So, it becomes: \[ 9 + 3 + 1 \]

Step 6: Simplify the Expression for Part (b)

Simplify the expression by performing the operations:\(9 + 3 = 12\).Now, \(12 + 1 = 13\).So, when \(x = -3\), the expression evaluates to 13.

Key Concepts

The Substitution Method

The substitution method in algebra is a straightforward way of solving or evaluating expressions. It essentially involves replacing a variable in an expression with a given number. This is especially useful in exercises that involve algebraic expressions like the example provided, where we need to evaluate the expression for specific values of variables.Here's how the substitution method works:

• Identify which variable in the expression you need to substitute. In our example, that variable is \( x \).
• Replace every occurrence of the variable with the given numerical value. For example, if \(x = 2\), replace every \(x\) in the expression with \(2\).
• Once you substitute the number, you'll see the new expression with numbers only. Now, it's ready to be simplified using other mathematical methods.

Always make sure the substitution is correctly done by consistently replacing all instances of the variable in the expression. This method is especially helpful in making abstract algebraic expressions more concrete and easier to handle.

The Order of Operations

Following the order of operations is crucial when simplifying expressions. It's a mathematically agreed-upon convention that tells us the sequence in which operations should be performed. The acronym PEMDAS can help remember the order:

• Parentheses first
• Exponents (or powers)
• Multiplication and Division (left to right)
• Addition and Subtraction (left to right)

For the expressions we evaluated, this rule ensures that we correctly simplify terms step by step, even when involving negative numbers or operations like subtraction from squared terms. Following PEMDAS simplifies the entire process:- Consider the operation inside any grouping symbols like parentheses first, if present.- Solve any exponents or powers next. For instance, in \((2)^2\), the exponent comes first.- Finally, perform any addition or subtraction from left to right. This diligence ensures clarity and correctness when simplifying expressions.

Evaluating Expressions

Evaluating expressions involves determining the numerical value of an expression once the variable is replaced with specific values. This concept combines both substitution and the order of operations to successfully arrive at the final result.Here's how we generally evaluate expressions:

• First, using substitution, replace variables with given numbers.
• Next, follow the mathematical order of operations to simplify the expression accurately.
• Finally, calculate step by step to find the expression’s value.

This process allows us to handle any algebraic expression, turning them into concrete numbers we can relate to. In the example exercise:- For \(x = 2\), substituting gives \((2)^2 - 2 + 1 = 3\).- For \(x = -3\), substituting yields \((-3)^2 - (-3) + 1 = 13\).Through systematic evaluation, expressions easily transform to reveal their intended numbers, helping solidify understanding for learners.