Problem 89 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 10. $$ b^{2}-4 a c \text { for } a=-1, b=5, \text { and } c=-2 $$

Step-by-Step Solution

Answer

The value of the expression is 17.

Step 1: Understand the Expression

We need to evaluate the expression \( b^2 - 4ac \) by substituting the given values for \( a \), \( b \), and \( c \). The values provided are \( a = -1 \), \( b = 5 \), and \( c = -2 \).

Step 2: Substitute Values into Expression

Replace \( a \), \( b \), and \( c \) in the expression \( b^2 - 4ac \) with the given values: \[ 5^2 - 4(-1)(-2) \]

Step 3: Calculate \( b^2 \)

Calculate \( b^2 \) by squaring \( b \): \( 5^2 = 25 \)

Step 4: Calculate \(-4ac\)

Calculate \(-4ac\) by multiplying: \[ -4(-1)(-2) = (-4) \times (-1) \times (-2) = -8 \].

Key Concepts

Substitution Method

The substitution method is like filling in the blanks for algebraic expressions. In our example with the quadratic expression \( b^2 - 4ac \), we replace the variables \( a \), \( b \), and \( c \) with the numbers given: \( a = -1 \), \( b = 5 \), and \( c = -2 \). This process is straightforward but essential because it transforms an abstract expression into a concrete calculation:

• Replace \( a \) with \(-1\).
• Replace \( b \) with \(5\).
• Replace \( c \) with \(-2\).

After substitution, the expression looks like this: \( 5^2 - 4(-1)(-2) \). This change means we can now solve the expression with numbers instead of variables, making it much easier to evaluate and understand.

Order of Operations

The order of operations determines which calculations you perform first in an expression. The standard order to solve any expression is remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication/division (from left to right)
• Addition/subtraction (from left to right)

In \( 5^2 - 4(-1)(-2) \), follow these steps:

• First, handle the exponent: calculate \( 5^2 \) which equals \( 25 \).


• Then multiply: do \(-4 \times (-1) \times (-2)\).


• Multiply sequentially: \(-4 \times (-1) = 4\), then \(4 \times (-2) = -8\).

Next, perform the subtraction: \( 25 + (-8) = 17 \). Adhering to these rules ensures that everyone solves expressions in the same way, and avoids confusion or errors.

Algebraic Simplification

Algebraic simplification is about making an expression as simple as possible. Here’s how we achieve this in our example. After substitution and following the order of operations, simplify step by step:

• First, perform the power operation: \(5^2 = 25\).
• Then, carry out the multiplication: \(-4 \times (-1) = 4\) and then \(4 \times (-2) = -8\).


• Finally, subtract to simplify: \(25 - 8 = 17\). This final step leaves us with a single number.

Simplification helps us to get to the simplest form of an expression, which is often a single value, much like reaching the bottom of a math puzzle. Carrying out these steps methodically helps to clarify the initial complexity, leading to clear solutions.