Chapter 10: Problem 48
Evaluate \(\frac{3 x^2+7 x-2}{2 x^2-7 x+3}\) for \(x=-4\) A. \(\frac{74}{7}\) B. \(\frac{18}{7}\) C. \(\frac{2}{7}\) D. 78
Short Answer
Expert verified
C. \( \frac{2}{7} \)
Step by step solution
01
- Substitute the value of x
Substitute the given value of \(x = -4\) into the expression: \[ \frac{3(-4)^2 + 7(-4) - 2}{2(-4)^2 - 7(-4) + 3} \]
02
- Simplify the numerator
Calculate the value of the numerator: \( 3(-4)^2 + 7(-4) - 2 = 3(16) + 7(-4) - 2 = 48 - 28 - 2 = 18 \)
03
- Simplify the denominator
Calculate the value of the denominator: \( 2(-4)^2 - 7(-4) + 3 = 2(16) + 7(4) + 3 = 32 + 28 + 3 = 63 \)
04
- Divide the simplified numerator by the simplified denominator
Divide the simplified numerator by the simplified denominator: \( \frac{18}{63} \)
05
- Simplify the fraction
Simplify the fraction \( \frac{18}{63} \): \( \frac{18 \times 1}{63 \times 1} = \frac{18}{63} = \frac{6 \times 3}{21 \times 3} = \frac{6}{21} = \frac{2}{7} \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
algebraic expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables (like x and y), and operators (addition, subtraction, multiplication, and division). This expression does not include an equality sign (=). For example, in the exercise, \(\frac{3x^2 + 7x - 2}{2x^2 - 7x + 3}\) is an algebraic expression with variables and constants.
When you're asked to evaluate such an expression, you need to replace the variable with a given number. This process is known as substitution. However, determining the exact value involves steps like simplifying the involved components: numerators and denominators. Being familiar with these will ease the process.
When you're asked to evaluate such an expression, you need to replace the variable with a given number. This process is known as substitution. However, determining the exact value involves steps like simplifying the involved components: numerators and denominators. Being familiar with these will ease the process.
substitution method
The substitution method is a crucial technique in algebra. It involves replacing the variable in an expression with a specific value. This step is essential to finding the numeric outcome of the expression.
In the given example, we were asked to evaluate the expression \(\frac{3x^2 + 7x - 2}{2x^2 - 7x + 3}\) for \(x = -4\).
In the given example, we were asked to evaluate the expression \(\frac{3x^2 + 7x - 2}{2x^2 - 7x + 3}\) for \(x = -4\).
- First, substitute \(x = -4\) in both the numerator and the denominator.
- Then, the numerator becomes \(3(-4)^2 + 7(-4) - 2\) and the denominator becomes \(2(-4)^2 - 7(-4) + 3\).
simplifying fractions
Simplifying fractions is about reducing them to their simplest form, where the numerator and denominator share no common factors other than 1. This process helps in making expressions cleaner and easier to interpret.
From the example provided, after substitution and mathematical operations, we reached the fraction \(\frac{18}{63}\).
From the example provided, after substitution and mathematical operations, we reached the fraction \(\frac{18}{63}\).
- To simplify, look for common factors. Both 18 and 63 are divisible by 3.
- Dividing the numerator and denominator by 3, we get \(\frac{18}{63} = \frac{6}{21}\).
- Repeating this with the new fraction, since 6 and 21 are both divisible by 3, we have \(\frac{6}{21} = \frac{2}{7}\).
GED test preparation
Preparing for the GED test requires understanding and mastering several mathematical concepts. The GED math reasoning section covers algebraic expressions, solving equations, and simplifying fractions, among other topics.
To excel:
To excel:
- Practice regularly on problems involving algebraic expressions. This will enhance your familiarity and problem-solving speed.
- Revisit key concepts like the substitution method, as it's commonly used in evaluations and solving equations.
- Work on simplifying fractions often, as it ensures clarity and correctness in final answers.