Problem 50 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(x^{2}-x+7\) for a. \(x=6\) b. \(x=-2\)

Step-by-Step Solution

Answer

For \( x=6 \), the expression evaluates to 37. For \( x=-2 \), it evaluates to 13.

Step 1: Substitute and evaluate for x=6

First, we substitute the given value of \( x = 6 \) into the expression \( x^2 - x + 7 \). This becomes \( 6^2 - 6 + 7 \). Calculate: \( 6^2 = 36 \), so the expression is now \( 36 - 6 + 7 \). Next, simplify: \( 36 - 6 = 30 \), and then \( 30 + 7 = 37 \). Thus, the value of the expression for \( x = 6 \) is 37.

Step 2: Substitute and evaluate for x=-2

Now, substitute \( x = -2 \) into the expression \( x^2 - x + 7 \). This gives us \( (-2)^2 - (-2) + 7 \). Calculate: \( (-2)^2 = 4 \), so the expression is \( 4 - (-2) + 7 \). Next, recognize that \( -(-2) = +2 \), thus it becomes \( 4 + 2 + 7 \). Finally, add these together: \( 4 + 2 = 6 \), then \( 6 + 7 = 13 \). Therefore, the value of the expression for \( x = -2 \) is 13.

Key Concepts

Substitution Method

The substitution method is an essential approach in algebra where you replace a variable in an expression with a given value. This makes the operation straightforward and allows for evaluating expressions accurately. To use this method:

• Take the algebraic expression and identify the variable you are substituting.
• Replace each instance of this variable with the given number.
• Follow the order of operations to simplify the expression and find the result.

For example, in the expression \( x^2 - x + 7 \), if we substitute \( x = 6 \), we re-calculate each term where \( x \) appears by replacing it with \( 6 \). This is an invaluable method for finding specific values of algebraic expressions, especially in polynomial expressions.

Evaluating Algebraic Expressions

Evaluating an algebraic expression involves finding the value of the expression for given variable values. It's a skill that's essential for solving many algebra problems. Here's how it typically works:

• First, substitute the given value for the variable into the expression.
• Next, perform the operations following the standard mathematical order: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Using the expression \( x^2 - x + 7 \), suppose \( x \) is given as \(-2\). By substituting and evaluating step-by-step, you can clearly see how each operation transforms the expression into a numerical answer. Evaluating algebraic expressions is straightforward but requires careful attention to detail to ensure accuracy.

Polynomial Expressions

Polynomial expressions are a key component in algebra and consist of coefficients, variables, and exponents combined through operations such as addition and subtraction. They can look like \( x^2 - x + 7 \), where this expression is a simple polynomial with a degree of 2 due to the highest power of \( x \), which is 2.Features of polynomial expressions include:

• Coefficients: Numbers multiplied by the variable, like \( 1x \) in \( -x \), where \( -1 \) is the coefficient.
• Terms: Each part of the expression separated by addition or subtraction; in \( x^2 - x + 7 \), there are three terms.
• Constant: A term without a variable, like \( 7 \) in this expression.

Polynomial expressions are fundamental in various mathematical calculations, from simple evaluations to complex calculus problems. Understanding their structure helps in simplifying, evaluating, and manipulating them to solve algebraic equations.