Question

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

C. \( \frac{2}{7} \)

Step-by-step solution

- Substitute the value of x

Substitute the given value of \(x = -4\) into the expression: \[ \frac{3(-4)^2 + 7(-4) - 2}{2(-4)^2 - 7(-4) + 3} \]

- Simplify the numerator

Calculate the value of the numerator: \( 3(-4)^2 + 7(-4) - 2 = 3(16) + 7(-4) - 2 = 48 - 28 - 2 = 18 \)

- Simplify the denominator

Calculate the value of the denominator: \( 2(-4)^2 - 7(-4) + 3 = 2(16) + 7(4) + 3 = 32 + 28 + 3 = 63 \)

- Divide the simplified numerator by the simplified denominator

Divide the simplified numerator by the simplified denominator: \( \frac{18}{63} \)

- Simplify the fraction

Simplify the fraction \( \frac{18}{63} \): \( \frac{18 \times 1}{63 \times 1} = \frac{18}{63} = \frac{6 \times 3}{21 \times 3} = \frac{6}{21} = \frac{2}{7} \)

Key concepts

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.

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 \(x = -4\).

• 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 \(2(-4)^2 - 7(-4) + 3\).

This method makes it manageable to handle complex expressions by breaking them down into simpler numeric calculations.

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 \(\frac{18}{63}\).

• 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 \(\frac{6}{21} = \frac{2}{7}\).

Simplifying makes the fraction much easier to understand and use in further calculations.

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:

• 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.

Utilizing practice tests and reviewing mistakes will significantly improve your performance and confidence.