Problem 13 - Fundamentals of Mathematics

Question

Find the value of each expression. $$(-x)^{2}+2 x+7, \text { if } x=4$$

Step-by-Step Solution

Answer

The value of the expression is 31.

Step 1: Substitute x with 4

Replace every occurrence of the variable \( x \) in the expression \((-x)^{2} + 2x + 7\) with \( 4 \). This gives us \((-4)^{2} + 2(4) + 7\).

Step 2: Evaluate the power

Calculate \((-4)^{2}\). Since squaring a negative number makes it positive, \((-4)^{2} = 16\).

Step 3: Calculate the multiplication

Multiply \(2\) by \(4\). Hence, \(2 \times 4 = 8\).

Step 4: Combine all terms

Add all the evaluated terms together. That is, compute \(16 + 8 + 7\).

Step 5: Final calculation

Evaluate \(16 + 8 + 7\) to get the final answer, which is \(31\).

Key Concepts

Substitution

In basic algebra, substitution involves replacing a variable in an expression with a given number or expression. It's like filling in the blanks. In the given exercise, we are asked to find the value of the expression \[ (-x)^2 + 2x + 7 \] when \( x = 4 \).
Here's how to proceed with substitution:

• Every time you see the variable \( x \) in the expression, swap it out for \( 4 \).
• This transforms the expression from \( (-x)^2 + 2x + 7 \) to \( (-4)^2 + 2(4) + 7 \).

Substitution simplifies the expression and allows you to evaluate it for a specific value. Always double-check that you've replaced each instance of the variable correctly.

Exponents

Exponents are a way to represent repeated multiplication. In our expression, we see an exponent in \((-4)^2\). This notation means that we multiply \(-4\) by itself.
Here's a simple breakdown:

• The number \(-4\) is being squared, which is written as \((-4)^2\).
• Squaring a number means multiplying the number by itself: \(-4 \times -4\).
• When you multiply two negative numbers, the result is positive, so \((-4)^2 = 16\).

Remember, raising a negative number to an even power results in a positive number. This principle is crucial for solving problems with exponents correctly.

Order of Operations

The order of operations is the rule that tells us the sequence to follow when performing mathematical operations, such as addition, subtraction, multiplication, and exponentiation.
To correctly evaluate the expression \((-4)^2 + 2(4) + 7\), follow these steps according to the order of operations:

• First, evaluate any exponents or powers. That's why we first calculated \((-4)^2 = 16\).
• Next, perform any multiplication. Here, \(2 \times 4 = 8\).
• Finally, carry out addition and subtraction from left to right. Thus, adding all terms gives us \(16 + 8 + 7\).
• This simplifies to a final value of \(31\).

The order of operations can be remembered with the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right). Always follow these rules to ensure you solve expressions correctly.