Problem 30 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ m^{2}-8 m-6, \text { if } m=-5 $$

Step-by-Step Solution

Answer

The value of the expression when \( m = -5 \) is 59.

Step 1: Substitute the value of m

The expression given is \( m^2 - 8m - 6 \). We are asked to find the value of this expression when \( m = -5 \). Begin by substituting \( -5 \) into the expression in place of \( m \). This gives us:\[ (-5)^2 - 8(-5) - 6 \]

Step 2: Calculate the square

Next, calculate the square of \(-5\). We have:\((-5)^2 = 25\). So, the expression now is:\[ 25 - 8(-5) - 6 \]

Step 3: Calculate the product of (-8) and (-5)

Calculate \(-8 \times -5\). This yields \( 40 \), since multiplying two negative numbers results in a positive number. Now the expression is:\[ 25 + 40 - 6 \]

Step 4: Perform the addition and subtraction

Now, add and subtract the numbers in the expression:First, do the addition: \( 25 + 40 = 65 \).Then, subtract \( 6 \) from \( 65 \), which gives:\[ 65 - 6 = 59 \]

Step 5: Final Result

After evaluating the expression, the final result is \( 59 \). Thus, \( m^2 - 8m - 6 \) for \( m = -5 \) is equal to \( 59 \).

Key Concepts

Substitution Method

The substitution method is a fundamental algebraic technique used to solve expressions and equations. It involves replacing a variable in an expression with a specific value to simplify and solve it. This approach is commonly used for evaluating polynomials and solving systems of equations.

To successfully apply the substitution method in our example, follow these steps:

• Identify the expression to be evaluated. In this case, the expression provided is \( m^2 - 8m - 6 \).
• Locate the value assigned to the variable. Here, we're given \( m = -5 \).
• Substitute the variable \( m \) with \( -5 \) throughout the expression. This transforms the expression into \( (-5)^2 - 8(-5) - 6 \).

Replacing variables with numbers simplifies complex problems into more straightforward arithmetic calculations. This step is crucial for making the expression manageable and to proceed with arithmetic operations.

Arithmetic Operations

Arithmetic operations form the basis of simplifying and solving algebraic expressions. They include addition, subtraction, multiplication, and division. Each operation follows specific rules, especially when handling polynomials, as seen in the given expression.

Initially, in the expression \( (-5)^2 - 8(-5) - 6 \), we must solve the squared term. Calculating \((-5)^2\) results in \( 25 \), as squaring any number involves multiplying the number by itself.

• Understand that squaring negates any sign, since \( (-5) \times (-5) = 25 \).

Next, we handle the multiplication occurring in \(-8(-5) \). Multiplying two negatives results in a positive, so \( -8 \times -5 \) yields \( 40 \). Now the expression is \( 25 + 40 - 6 \).

Conclude with straightforward addition and subtraction:

• Add systematically: \( 25 + 40 = 65 \)
• Then subtract: \( 65 - 6 = 59 \)

These steps use basic arithmetic to simplify and solve the expression fully, providing confidence in your ability to handle more complex polynomial tasks.

Negative Numbers Handling

Understanding how to manage negative numbers is crucial in algebra, especially when dealing with polynomials. Negative numbers follow specific rules in arithmetic operations that are pivotal for avoiding errors in calculations.

There are a few key principles when working with negative numbers:

• When you square a negative number, the result is positive. For example, \( (-5)^2 = 25 \).
• Multiplying two negative numbers together yields a positive result, which is why \( -8 \times -5 = 40 \).
• Adding and subtracting negative numbers might seem tricky, but remember: subtracting a negative is the same as adding a positive. This simplifies scenarios like "\(-8 \times (-5)\) leading to a positive 40."

Having these concepts clear can transform challenging polynomial expressions into simpler tasks. Negative numbers often trip people up, but with practice, handling them can become second nature. With a strong grasp on these basics, your calculations will be more accurate and less prone to mistakes.