Problem 43 - Elementary Algebra

Question

Evaluate \(2(x+h) 2-5(x+h)+3,\) given \(x=2\) and \(h=0.1\)

Step-by-Step Solution

Answer

The expression evaluates to 1.32.

Step 1: Substitute the Values

Start by substituting the given values of \(x = 2\) and \(h = 0.1\) into the expression \(2(x+h)^2 - 5(x+h) + 3\). This results in the expression \(2(2+0.1)^2 - 5(2+0.1) + 3\).

Step 2: Simplify Inside the Parentheses

Add \(x\) and \(h\) inside the parentheses to get \(2.1\). So, the expression becomes \(2(2.1)^2 - 5(2.1) + 3\).

Step 3: Square the Term

Calculate \((2.1)^2\), which equals \(4.41\). Substitute back into the expression to get \(2(4.41) - 5(2.1) + 3\).

Step 4: Multiply the Squared Term

Multiply \(2\) and \(4.41\) to get \(8.82\). The expression now reads \(8.82 - 5(2.1) + 3\).

Step 5: Multiply the Linear Term

Multiply \(5\) and \(2.1\) to get \(10.5\). Now the expression becomes \(8.82 - 10.5 + 3\).

Step 6: Combine the Terms

Subtract \(10.5\) from \(8.82\) to get \(-1.68\) and then add \(3\) to \(-1.68\) to arrive at the final result. The final result is \(1.32\).

Key Concepts

Evaluating Expressions

Evaluating algebraic expressions involves finding the value of the expression by substituting numbers for the variables. This is a fundamental skill in algebra. When given an expression and specific values for the variables, you perform operations to simplify and arrive at a numerical result.

In the given exercise, you begin with the expression \(2(x+h)^2 - 5(x+h) + 3\). The values provided are \(x = 2\) and \(h = 0.1\). Start by replacing these variables with the given numbers, transforming the expression into a numeric form:

• Substitute \(x = 2\) and \(h = 0.1\)
• The expression becomes \(2(2+0.1)^2 - 5(2+0.1) + 3\)

This way, you're ready to perform arithmetic operations to simplify and determine the expression's numerical value.

Substitution in Algebra

Substitution is a key technique in algebra that involves replacing variables in an expression with their actual values or numbers. This process helps in evaluating the expression to find its numerical value.

Here's a simple way to substitute:

• Ensure you clearly know the values you are substituting. In our example: \(x = 2\) and \(h = 0.1\).
• Replace every occurrence of the variables \(x\) and \(h\) with these numbers.

This results in an expression, like \(2(2+0.1)^2 - 5(2+0.1) + 3\), where you have specific numbers instead of variables.

After substitution, proceed with operational steps such as simplifying within parentheses, performing basic arithmetic operations, like squaring numbers, multiplying, and combining terms.

Simplifying Algebraic Expressions

Simplifying expressions involves reducing them to their simplest form, making calculations easier and clearer. Here's how you can simplify the expression from the given problem:

• First, solve inside the parentheses. Add \(2\) and \(0.1\) to get \(2.1\).
• Next, square the result: \((2.1)^2 = 4.41\).
• Then, follow the order of operations, multiplying and subtracting the results: \(2 \times 4.41 = 8.82\) and \(5 \times 2.1 = 10.5\).
• Subtract \(10.5\) from \(8.82\) to get \(-1.68\) and finally, add \(3\) to result in \(1.32\).

By completing these steps, you have taken a complex expression and simplified it down to a single number. Simplifying is crucial in algebra, as it ensures your final answer is concise and manageable.