Problem 8 - Fundamentals of Mathematics

Question

Find the value of each expression. $$ -a^{2}+3 a+4, \text { if } a=4. $$

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Identify the Expression

We are given the algebraic expression \(-a^2 + 3a + 4\). We need to find its value when \(a = 4\).

Step 2: Substitute the Given Value

Replace \(a\) with 4 in the expression. This gives us:\(-4^2 + 3(4) + 4\).

Step 3: Evaluate the Exponents

Calculate \(4^2\). This equals \(16\). Now the expression is:\(-16 + 3(4) + 4\).

Step 4: Perform Multiplication

Multiply \(3\) by \(4\), which gives \(12\). The expression is now:\(-16 + 12 + 4\).

Step 5: Perform Addition and Subtraction

First, add \(12\) and \(4\) to get \(16\). The expression becomes:\(-16 + 16\).

Step 6: Simplify the Expression

Subtract \(16\) from \(16\) to get \(0\). Thus, the value of the expression when \(a = 4\) is \(0\).

Key Concepts

Substitution Method

The substitution method involves replacing a variable in an algebraic expression with a given value. This process is fundamental in solving expressions when specific values are provided, like in our example where we substitute \( a = 4 \) into the expression \(-a^2 + 3a + 4\).

Here’s how substitution streamlines problem-solving:

• Replace Variables: In the expression \(-a^2 + 3a + 4\), replace each occurrence of \( a \) with \( 4 \), resulting in \(-4^2 + 3(4) + 4\).
• Solve Simplified Expression: With all values in place, solve the equation step-by-step to get the final result.

Substitution transforms abstract algebraic expressions into arithmetic ones that are more straightforward to solve. This is essential for evaluating expressions at particular points.

In our example, by substituting the known value \( a = 4 \), the expression became a series of numbers to be summarized, thus leading to the final solution.

Exponents

Exponents are a way of representing repeated multiplication of a number by itself. In our exercise, we encounter \(-a^2\) where the exponent \(2\) indicates that the variable \(a\) is multiplied by itself.

The process for handling exponents is straightforward:

• Identify the Base: The base is \(a\), which equals \(4\) in this case.
• Apply the Exponent: Calculate \(4^2\), which is \(4 \times 4 = 16\).
• Consider Negative Signs: In \(-a^2\), the negative sign applies to the result of \(a^2\), not the base itself.

So, \(-a^2\) becomes \(-16\) when \(a = 4\). Understanding how to compute exponents correctly is vital as they can significantly change the outcome of your calculations.

Exponents reduce complex multiplications into manageable values and are vital for concise math expressions, particularly in algebra.

Order of Operations

The order of operations is a set of rules that dictate the sequence in which various operations within an expression should be carried out to ensure consistent results, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Applying this to our example:

• Evaluate Exponents: Begin with the exponent, calculating \(4^2\) to get \(16\).
• Perform Multiplications and Divisions: Next, handle multiplication, which includes multiplying \(3 \times 4\) to obtain \(12\).
• Addition and Subtraction: Finally, add and subtract the results sequentially: \(-16 + 12 + 4 = 0\).

By strictly following the order of operations, we ensure each step is executed in the proper sequence, which maintains the integrity of the mathematical process.

Neglecting these rules may lead to incorrect answers, making it crucial to discipline oneself in adhering to these standards.