Problem 23 - Elementary Algebra

Question

For each problem below, evaluate \(b_{2}-4 a c\), given the following values for \(a, b\), and \(c\). $$ a=1, b=2, c=3 $$

Step-by-Step Solution

Answer

The value is \(-8\).

Step 1: Identify the Expression

The expression given in the problem is \( b^2 - 4ac \). Our task is to evaluate this expression using the provided values of \( a \), \( b \), and \( c \).

Step 2: Substitute the Given Values

Insert the given values into the expression: \( b = 2 \), \( a = 1 \), and \( c = 3 \). Now, the expression becomes:\[ 2^2 - 4 \times 1 \times 3 \]

Step 3: Calculate the Power of b

First, calculate \( b^2 \). Since \( b = 2 \), we have:\[ 2^2 = 4 \]

Step 4: Calculate the Product of 4ac

Next, compute \( 4ac \). Substitute the values \( a = 1 \) and \( c = 3 \):\[ 4 \times 1 \times 3 = 12 \]

Step 5: Compute the Final Expression

Subtract the result of \( 4ac \) from \( b^2 \):\[ 4 - 12 = -8 \]

Step 6: Conclusion

The result of the expression \( b^2 - 4ac \) using the given values is \(-8\).

Key Concepts

Substitution in Expressions

In algebra, substitution is a fundamental technique used to simplify and solve expressions and equations. It involves replacing variables with given numbers to evaluate the expression. For example, consider the expression \( b^2 - 4ac \).
To substitute, we replace:

• \( b \) with 2,
• \( a \) with 1, and
• \( c \) with 3.

Now, the expression becomes \( 2^2 - 4 \times 1 \times 3 \). This step is crucial because it sets up the problem for further calculations.
By substituting the values, you transform an abstract expression into a concrete one, making it easier to work with. Understanding this concept allows you to tackle similar problems with confidence.

Evaluating Expressions

Evaluating an expression means simplifying it to find its numerical value, using the order of operations. When we have our substituted expression \( 2^2 - 4 \times 1 \times 3 \), it's time to simplify it.
First, according to the order of operations (often remembered by PEMDAS/BODMAS: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), start with exponents.

• Calculate \( 2^2 \), which equals 4.
• Next, compute the multiplication \( 4 \times 1 \times 3 \), resulting in 12.

Finally, subtract \( 4 \) from \( 12 \), giving \( -8 \).
Evaluating expressions accurately depends on understanding and applying the correct order of operations. Practicing this method will simplify complex expressions effectively.

Algebraic Calculations

Algebraic calculations involve performing operations on mathematical expressions to simplify or solve them. With our expression \( b^2 - 4ac \), algebra helps us understand the role of operations like exponents, multiplication, and subtraction.
A step-by-step approach to algebraic calculations helps in achieving accurate results:

• Perform exponents first: In our case, computing \( 2^2 \) results in 4.
• Focus on multiplication: The term \( 4 \times 1 \times 3 \) calculates to 12.
• Lastly, handle addition or subtraction: Subtracting 12 from 4 yields a result of \( -8 \).

Engaging with algebraic calculations teaches you to handle problems systematically. It's all about breaking down expressions into manageable parts and applying basic arithmetic operations.