Problem 29 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ a^{2}-6 a+4, \text { if } a=-2 $$

Step-by-Step Solution

Answer

The value of the expression is 20.

Step 1: Substitute the value of 'a'

The given expression is \(a^2 - 6a + 4\). We need to substitute \(a = -2\) into the expression. This gives us \((-2)^2 - 6(-2) + 4\).

Step 2: Simplify the squared term

Calculate \((-2)^2\). Since squaring a negative number results in a positive number, \((-2)^2\) is equal to \(4\).

Step 3: Simplify the product term

Calculate \(-6(-2)\). Multiplying two negative numbers yields a positive number, so \(-6 \times -2 = 12\).

Step 4: Combine all terms

Substitute the results from Steps 2 and 3 back into the expression, resulting in \(4 + 12 + 4\).

Step 5: Calculate the final result

Add the numbers together: \(4 + 12 + 4 = 20\).

Key Concepts

Substitution

Substitution is an essential technique when working with algebraic expressions. It involves replacing a variable in an expression with a given number. In our exercise, the expression is \(a^2 - 6a + 4\) and we are told to find its value when \(a = -2\). To substitute, simply replace every instance of the variable \(a\) in the expression with \(-2\). Thus, the expression becomes:

• \((-2)^2\) for \(a^2\)
• \(-6(-2)\) for \(-6a\)
• The constant \(4\) remains the same.

Substitution sets the stage for further simplification and calculations.

Simplification

Simplification is the process of reducing an expression to its simplest form, making it easier to evaluate. Once we have substituted \(a = -2\) into \(a^2 - 6a + 4\), we simplify the expression to make calculations straightforward.For our example, after substitution, we have:

• \((-2)^2 - 6(-2) + 4\)

Breaking it down:

• First, calculate \((-2)^2\).
• Second, handle the multiplication \(-6(-2)\).
• Finally, add the constant \(4\).

Simplification reduces each term individually so the entire expression is easy to compute.

Squaring Numbers

Squaring a number means multiplying it by itself. When you square a number, the result is always non-negative, because multiplying two negatives gives a positive and multiplying two positives gives a positive.For our substitution of \(a = -2\):

• Consider \((-2)^2\).
• This translates to \((-2) \times (-2)\).
• Therefore, \((-2)^2 = 4\).

Squaring negative numbers follows the same rule: the squared term becomes positive. This step is crucial in ensuring the subsequent calculations remain correct.

Multiplying Negative Numbers

Multiplying negative numbers can sometimes be tricky, but there's a simple rule: the product of two negative numbers is positive. This is important in algebraic expressions where signs can change the value of the entire expression. For our expression, after substituting \(a = -2\) in the term \(-6a\), we have:

• Calculate \(-6(-2)\).
• The multiplication becomes \(-6 \times -2\).
• Using the rule, \(-6(-2) = 12\).

Understanding this concept is vital as it determines the overall value when combined with other terms in the expression to reach the final result.