Problem 42 - Algebra 1

Question

Evaluate the expression. $$b^{2}-4 a c \text { when } a=1, b=3, \text { and } c=5$$

Step-by-Step Solution

Answer

-11

Step 1: Substitute the given values

Substitute \(a = 1\), \(b = 3\), and \(c = 5\) into the given expression \(b^{2} - 4ac\). The new expression becomes \(3^{2} - 4 \cdot 1 \cdot 5\).

Step 2: Perform the operations

Perform the operations in the correct order according to the BIDMAS/BODMAS rule (Brackets, Indices/Orders, Division, Multiplication, Addition, Subtraction). We first conduct the exponentiation operation \(3^{2}\) giving us 9, followed by the multiplication \(4 \cdot 1 \cdot 5\), yielding 20. The expression now reads as \(9 - 20\).

Step 3: Final calculation

We finish by performing the subtraction operation, giving us a final result of -11.

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(a\) or \(b\)), and operation symbols (such as +, −, ×, ÷). They do not have an equal sign; this distinguishes them from equations.
In the exercise provided, the algebraic expression is \(b^2 - 4ac\). Each part of this expression has its unique role.

• \(b^2\): This is \(b\) raised to the power of 2, indicating a multiplication of \(b\) by itself.
• \(4ac\): This part involves the multiplication of three different terms: 4, \(a\), and \(c\).
• Subtraction (−): The subtraction sign separates these terms, showing that we subtract the product \(4ac\) from the square of \(b\).

Understanding algebraic expressions is crucial as they are foundational in all branches of mathematics, from simple calculations to complex equations.

BIDMAS/BODMAS Rule

BIDMAS or BODMAS is an acronym that helps us remember the order of operations in mathematics: Brackets, Indices (Orders), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This rule ensures we perform calculations in the correct sequence to arrive at the right answer.
In the given problem, after substituting in the values, our expression is \(3^2 - 4 \times 1 \times 5\). The BIDMAS rule guides us to:

• First handle any Brackets, but there are none in this expression.
• Next, evaluate Indices, which is \(3^2 = 9\).
• Then perform the Multiplication: \(4 \times 1 \times 5 = 20\).
• Lastly, perform the Subtraction: \(9 - 20 = -11\).

Following BIDMAS/BODMAS ensures that expressions are evaluated correctly.

Substitution Method

The substitution method involves replacing variables in an expression with specific values. This technique allows us to evaluate expressions by turning them into simpler arithmetic.
In our exercise, we're given specific values for each variable: \(a = 1\), \(b = 3\), \(c = 5\).

• Start by identifying the variables in the expression \(b^2 - 4ac\).
• Replace each with its given value: substitute \(b = 3\), \(a = 1\), and \(c = 5\) to transform the expression.
  
• The expression becomes \(3^2 - 4 \times 1 \times 5\).

This substitution turns what might seem like a complex algebraic task into a straightforward numerical calculation. It's a valuable skill for solving algebraic problems.

Order of Operations

The order of operations is fundamental to accurately solving mathematical expressions. It's easy to think of it as "the rules" for math.
The order of operations ensures consistency and accuracy:

• Brackets first
• Indices (powers or roots, like squares or square roots)
• Then we handle Division and Multiplication from left to right
• Finally, Addition and Subtraction from left to right

For our specific problem, once we substituted \(a = 1\), \(b = 3\), and \(c = 5\) into \(b^2 - 4ac\), we then applied these rules:

• Calculated \(3^2\) first, because it's an index.
• Followed by multiplying \(4 \times 1 \times 5\), which falls under multiplication.
• Lastly subtracted to get \(9 - 20\), yielding \(-11\).

By adhering to this order, complex expressions can be simplified and solved correctly every time.