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Chapter 1: Problem 75

Evaluate each algebraic expression for the given value of the variable. $$3 x^{2}-8 x ; x=-2$$

Short Answer

Expert verified
The evaluated result of the algebraic expression, \(3x^2 - 8x\), for \(x = -2\) is \(28\).

Step by step solution

01

Substitution

Replace \(x\) in the algebraic expression \(3x^2 - 8x\) with \(x = -2\). This gives us \(3(-2)^2 - 8(-2)\)
02

Simplification

First, calculate the value of \((-2)^2\), which equals \(4\), and substitute it back into the expression, which gives us: \(3 * 4 - 8(-2)\). Next, multiply \(3 * 4 = 12\) and \(-8*(-2) = 16\). This simplifies the new expression to \(12 + 16\).
03

Final Calculation

Now, calculate the final expression: \(12 + 16 = 28\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Substitution
Variable substitution is a fundamental concept in algebra that involves replacing a variable with a given number. This process helps in evaluating expressions, making it a key skill for solving algebraic problems.
In the exercise, we were given the expression \(3x^2 - 8x\) and asked to evaluate it for \(x = -2\). By substituting \(x = -2\) into the expression, we transform it into a numerical form: \(3(-2)^2 - 8(-2)\).
  • The essence of substitution is straightforward, yet it is crucial to carefully handle any negative values or fractions.
  • Ensure all occurrences of the variable in the expression are replaced consistently.
By doing this correctly, you set up the expression for further evaluation and simplification.
Polynomial Evaluation
Polynomial evaluation involves computing the value of a polynomial expression once a specific value for the variable is provided.
The aim is to determine the numerical outcome of the expression through arithmetic calculations.
For our specific example:
  • Start with the calculation of powers and products. Here, compute \((-2)^2\), resulting in \(4\).
  • This transforms our expression to \(3 \times 4 - 8 \times (-2)\).
Next, resolve the products:
  • Multiply \(3\) and \(4\) to get \(12\).
  • Calculate \(-8 \times (-2)\) to receive \(16\).
This step-by-step breakdown is essential in polynomial evaluation as it sets the stage for simplifying the expression further.
Expression Simplification
Simplifying an expression means combining like terms and following arithmetic rules to reduce the expression to its simplest form.
This is the final stage where all calculation parts come together.
After evaluating the individual parts of the polynomial, we get \(12 + 16\).
  • By adding these two results, we conclude with \(28\) as the final answer.
Expression simplification requires attention to detail, ensuring all arithmetic operations, such as addition, subtraction, multiplication, or division, are executed correctly. It leads to the most compact and clear form of the expression.

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