Problem 96 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ \frac{-b^{2}+16 a^{2}+1}{2} \text { for } a=\frac{1}{4} \text { and } b=-10 $$

Step-by-Step Solution

Answer

The expression evaluates to -49.

Step 1: Substitute the Given Values into the Expression

We start by substituting the given values of \( a = \frac{1}{4} \) and \( b = -10 \) into the expression. The original expression is:\[\frac{-b^{2}+16 a^{2}+1}{2}\]Plug the values in:\[\frac{-(-10)^{2}+16 \left(\frac{1}{4}\right)^{2}+1}{2}\]

Step 2: Calculate \( b^{2} \) and \( 16 a^{2} \)

First, calculate \( b^2 \):\((-10)^2 = 100\).Next, calculate \( 16a^2 \):\[16 \left(\frac{1}{4}\right)^{2} = 16 \times \frac{1}{16} = 1\]

Step 3: Plug the Calculated Values Back into the Expression

Now, substitute the calculated values into the expression:\[\frac{-100 + 1 + 1}{2}\]

Step 4: Simplify the Numerator

Simplify the terms in the numerator:\[-100 + 1 + 1 = -98\]

Step 5: Perform the Division

Finally, divide the simplified numerator by 2:\[\frac{-98}{2} = -49\]

Key Concepts

Evaluation of Expressions

Evaluating expressions involves calculating the value of an algebraic expression when the variables are replaced by their given numerical values. This process is key in algebra as it helps us understand how different values affect the outcome of the expression. The exercise example demonstrates evaluating the expression \( \frac{-b^{2}+16 a^{2}+1}{2} \) where specific values for \( a \) and \( b \) are provided.
When we evaluate an expression:

• We begin by substituting the given values into the expression.
• Next, we follow the order of operations, which is often remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
• Finally, we simplify the expression step by step to find the final value.

By evaluating expressions, we translate abstract algebraic expressions into concrete numerical results, making it a fundamental skill in mathematical problem-solving.

Substitution in Algebra

Substitution is an essential technique in algebra where specific values are assigned to variables in an expression. This step is critical when evaluating expressions because it transforms an abstract algebraic equation into a series of arithmetic operations that can be calculated.
In our exercise, substitution takes place at the start. Given \( a = \frac{1}{4} \) and \( b = -10 \), these values are "plugged" into our original expression formula \( \frac{-b^{2}+16 a^{2}+1}{2} \).
To substitute:

• Ensure each variable in the expression is replaced by its given value.
• Write the expression clearly, altering only the parts where the variables are substituted.

Substitution is straightforward once you recognize which variable corresponds to which value. Accurate substitution sets the foundation for correctly solving the overall expression.

Basic Arithmetic Operations

Basic arithmetic operations are the fundamental building blocks of mathematical calculations. These operations include addition, subtraction, multiplication, and division. When working with expressions, especially after substitution, these operations are systematically applied to compute the final value.
In the step-by-step solution for our exercise, several basic operations are employed:

• Compute \( b^2 \) by squaring the given value of \( b \). Here, \((-10)^2 = 100\).
• Calculate \( 16a^2 \) by first squaring \( a \), i.e., \( \left(\frac{1}{4}\right)^2 = \frac{1}{16} \), and then multiplying by 16, resulting in 1.
• Apply addition and subtraction in the numerator to combine terms like \(-100 + 1 + 1 = -98\).
• Finally, perform division by dividing the simplified numerator, \(-98\), by 2 to arrive at \(-49\).

Being comfortable with these basic operations is vital as they are often combined in various sequences to resolve different parts of an expression. Mastery of arithmetic lays the groundwork for tackling more complex algebraic challenges.