Problem 56 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(-10.3 x^{2}-x+6.5\) for a. \(x=-1\) b. \(x=-2\)

Step-by-Step Solution

Answer

For x = -1, the expression evaluates to -2.8; for x = -2, it evaluates to -32.7.

Step 1: Substitute for x in part (a)

We need to evaluate the expression for \( x = -1 \). Start by substituting \( -1 \) for \( x \) in the expression: -10.3(-1)^2 - (-1) + 6.5.

Step 2: Solve the expression for x = -1

Calculate each part:- \((-1)^2 = 1\) so \(-10.3 \times 1 = -10.3\)- The expression becomes \(-10.3 + 1 + 6.5\)- Calculate \(-10.3 + 1 = -9.3\)- Calculate \(-9.3 + 6.5 = -2.8\)Thus, the value is \(-2.8\).

Step 3: Substitute for x in part (b)

We now evaluate the expression for \( x = -2 \). Substitute \( -2 \) for \( x \): - 10.3(-2)^2 - (-2) + 6.5.

Step 4: Solve the expression for x = -2

Calculate each part:- \((-2)^2 = 4\) so \(-10.3 \times 4 = -41.2\)- The expression becomes \(-41.2 + 2 + 6.5\)- Calculate \(-41.2 + 2 = -39.2\)- Calculate \(-39.2 + 6.5 = -32.7\)Thus, the value is \(-32.7\).

Key Concepts

Substitution Method

The substitution method is like giving a value to a placeholder. Imagine you're cooking and need to swap an ingredient in a recipe. You substitute something in place of what was originally there.

In algebra, we do this by replacing variables with numbers. Consider the expression "\(-10.3x^{2}-x+6.5\)". We're asked to find out its value for different \(x\) values.

Here's how you do it:

• Choose the value for the variable, in this case, \(x = -1\) or \(x = -2\).
• Substitute this chosen value into every place where the variable \(x\) appears in the expression.

This simple process helps in turning an algebraic expression into a calculable mathematical problem. It's like giving each unknown a face, making the problem easier to handle.

Polynomial Evaluation

Evaluating a polynomial is like breaking down a recipe step by step. Each term uses the given variable's value, and you compute accordingly.

Let's look at our polynomial \(-10.3x^{2}-x+6.5\). Each part of this expression stands as a different ingredient:

• \(-10.3x^{2}\): Represents the variable squared multiplied by \(-10.3\).
• \(-x\): The variable itself, but its value is flipped to negative.
• \(+6.5\): A constant term added to the whole expression.

When you evaluate this for a particular \(x\) value, you:

• Calculate each term with the substituted number.
• Combine (add or subtract) these results.

Typically, start with the term involving the highest power of \(x\) and proceed to the constant. This is crucial for getting the right answer, much like adding ingredients in the correct order during cooking.

Order of Operations

The order of operations is like having a roadmap for solving math problems. It tells you which steps to take first and ensures that everyone solves math problems the same way. Remember the acronym PEMDAS to guide you:

• P for Parentheses: Solve expressions inside parentheses first.
• E for Exponents: Next, calculate powers or roots.
• M and D for Multiplication and Division: Do these from left to right.
• A and S for Addition and Subtraction: Lastly, do these from left to right.

With our expression \(-10.3x^{2}-x+6.5\), follow this order to evaluate correctly.

• First, apply the exponent (squaring the \(x\) value).
• Then, do the multiplication and handle addition or subtraction last.

By following these steps, you ensure your calculations are accurate and consistent, just like following the rules of a game.