Problem 73 - Algebra 1: Concepts and Skills

Question

Evaluate the expression when \(x=-2\) (Lessons \(1.3,2.3,2.5)\). $$ 3 x^{2}+4 x+8 $$

Step-by-Step Solution

Answer

The expression evaluates to 12 when \(x = -2\).

Step 1: Substitution

Substitute the value of x in the expression. Replace \(x\) with \(-2\) in the expression \(3x^{2} + 4x + 8\). That changes the expression to \(3*(-2)^{2} + 4*(-2) + 8\).

Step 2: Simplify the Expression

Simplify the expression using the order of operations, also known as BIDMAS/BODMAS (Brackets/Orders, Indices, Division/Multiplication, Addition/Subtraction). Firstly, \(3*(-2)^{2}\) calculates to \(3*4 = 12\). Then \(-4*2 = -8\). Add these together with 8, we get \(12 - 8 + 8\).

Step 3: Final Simplification

On simplifying the expression, we get \(12 - 8 + 8 = 12\).

Key Concepts

Substitution in Algebra

A key component in evaluating algebraic expressions is the concept of substitution. This involves replacing a variable with a given value. Algebra often uses letters, like \( x \), to stand for numbers in expressions.To solve these expressions, you follow a few simple steps:

• Identify the variable(s) in the expression. In our example, the variable is \( x \).
• Receive or choose a specific number for each variable. Here, we substitute \( x = -2 \).
• Replace every instance of the variable with its corresponding number. In the equation \( 3x^2 + 4x + 8 \), replace \( x \) with \( -2 \) to get \( 3(-2)^2 + 4(-2) + 8 \).

Substitution makes complex expressions easier to evaluate by turning them into numerical operations that can be solved step-by-step.

Order of Operations

When simplifying algebraic expressions with substitutions, it's crucial to follow the order of operations. This ensures we solve mathematical expressions correctly. The order is often remembered by the acronym BIDMAS/BODMAS, which stands for:

• Brackets
• Indices (or Orders, referring to powers and roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Following this order ensures that expressions are simplified systematically and accurately. For instance, in the expression \( 3(-2)^2 + 4(-2) + 8 \):
- We first calculate \((-2)^2\), which is \(4\), using the 'Indices' step.
- Then, multiply the square by \(3\) to get \(12\).
- Next, compute \(4(-2)\) to get \(-8\).
Finally, you finish by adding and subtracting the results in order: \(12 - 8 + 8 = 12\). Using BIDMAS/BODMAS keeps calculations organized and correct.

Simplification of Expressions

Simplification is the process of reducing expressions to their simplest form. By simplifying an expression, you make it easier to understand and work with. After substituting values and following the order of operations, the expression can usually be simplified further.Here’s how the simplification works in practice:

• Perform operations in the correct sequence, using the order of operations as a guide.
• Combine like terms if possible. This involves grouping and simplifying terms that have the same variables raised to the same powers.

In the original example, once we substitute \( x = -2 \), we follow BIDMAS/BODMAS to calculate:
1. \( 3(-2)^2 = 12 \)
2. \( 4(-2) = -8 \)
3. Add them: \( 12 - 8 + 8 \), which simplifies to \( 12 \).Simplification cuts through complex expression layers, providing a clear, accurate result.