Problem 16 - Beginning and Intermediate Algebra

Question

Evaluate each expression for ( \(\boldsymbol{a}\) ) \(x=4\) and \((\boldsymbol{b}) x=6\). \(5 x^{2}\)

Step-by-Step Solution

Answer

For \(x=4\), the result is 80. For \(x=6\), the result is 180.

Step 1: Understand the expression

The given expression is a mathematical formula: \(5x^2\). This means we need to multiply 5 by the square of \(x\).

Step 1a: Substitute \(x = 4\)

First, substitute \(x = 4\) into the expression: \(5(4)^2\).

Step 1b: Calculate \(4^2\)

Next, calculate the square of 4: \(4^2 = 16\).

Step 1c: Multiply by 5

Finally, multiply 16 by 5: \(5 \times 16 = 80\). Thus, the expression evaluated for \(x = 4\) is 80.

Step 2a: Substitute \(x = 6\)

Now, substitute \(x = 6\) into the expression: \(5(6)^2\).

Step 2b: Calculate \(6^2\)

Next, calculate the square of 6: \(6^2 = 36\).

Step 2c: Multiply by 5

Finally, multiply 36 by 5: \(5 \times 36 = 180\). Thus, the expression evaluated for \(x = 6\) is 180.

Key Concepts

Substitution

Substitution is one of the most fundamental concepts in algebra. It involves replacing a variable with a given number or value. For example, in the expression \(5x^2\), if we are given \(x = 4\), we substitute 4 wherever there is an x in the expression. So, \(5x^2\) becomes \(5(4)^2\). This method helps us convert algebraic expressions into numerical computations, making it easier to evaluate them.

Here are some important points to consider for substitution:

• Identify the variable in the expression.
• Replace every occurrence of the variable with the given number.
• Ensure that you maintain the structure of the expression while substituting.

Substitution simplifies the problem into something more manageable, allowing you to proceed to the next calculation steps such as squaring or multiplying.

Squaring a Number

Squaring a number means multiplying that number by itself. It is a basic operation in algebra. For instance, squaring 4 gives us \(4 \times 4 = 16\). In our expression \(5x^2\), once we substitute \(x\) with 4, we need to square the 4:

\( (4)^2 = 16 \).

This is because any number raised to the power of 2 is multiplied by itself. Another example is squaring 6:

\((6)^2 = 36\).

The squaring operation is crucial when dealing with quadratic expressions. Remember these key points about squaring:

• Ensure you multiply the number by itself, not by any other number.
• Squaring a negative number will result in a positive number, as multiplying two negative numbers yields a positive.
• Double-check your calculations to avoid errors.

Understanding how to square numbers accurately is essential for correctly evaluating algebraic expressions.

Multiplication

Multiplication is the arithmetic operation of scaling one number by another. After substituting the variable and squaring the number in the expression \(5x^2\), the next step involves multiplication. For example, once we have \(4^2 = 16\), in the expression \(5(4)^2\), we need to multiply 5 by 16:

\(5 \times 16 = 80\).

This operation combines two quantities to produce a product. Let's see another example where \(x = 6\):

First, square the 6:

\(6^2 = 36\).

Then, multiply by 5:

\(5 \times 36 = 180\).

Key points about multiplication to keep in mind:

• Ensure you are multiplying the correct numbers in sequence.
• Understand that multiplication is associative, meaning \((a \times b) = (b \times a)\).
• Use multiplication properties, such as distributive property, for more complex expressions.

Multiplication solidifies the final solution of the expression, making it possible to complete the evaluation.