Problem 36 - Algebra 2

Question

A parallelogram with an area of \(6 x^{2}-7 x-5\) square units has a base of \(3 x-5\) units. Determine the height of the parallelogram.

Step-by-Step Solution

Answer

The height of the parallelogram is \(2x + 1\) units.

Step 1: Understand the Formula for Area of a Parallelogram

The area of a parallelogram can be calculated using the formula \( \text{Area} = \text{base} \times \text{height} \). In this problem, we are given the area as \(6x^2 - 7x - 5\) and the base as \(3x - 5\). We need to find the height \(h\).

Step 2: Substitute Known Values into the Formula

Substitute the given expressions into the formula:\[6x^2 - 7x - 5 = (3x - 5) \times h.\]

Step 3: Solve for the Height

To find the height \(h\), solve the equation by isolating \(h\):\[h = \frac{6x^2 - 7x - 5}{3x - 5}.\]

Step 4: Simplify the Expression

To ensure the expression is fully simplified, use polynomial division to divide \(6x^2 - 7x - 5\) by \(3x - 5\). After performing the division, we find that:\[h = 2x + 1.\] (Ensure to double-check the division and any remainder is managed properly.)

Key Concepts

Parallelogram Area Formula

The parallelogram area formula is essential in geometry, allowing you to calculate the area of a parallelogram using its base and height. The formula itself is straightforward:

• Area = Base × Height

For a given parallelogram, if you know the length of the base and the height, you multiply these two values together to find the area. This formula reflects the idea that if you can visualize a parallelogram as a stretched rectangle, the height effectively becomes the "breadth" of that rectangle.
In practical problems, such as our exercise, this formula is used inversely. We start with the area and one dimension (the base) to determine the unknown dimension (the height). Recognizing and understanding how to manipulate this formula in different scenarios is crucial for solving real-world problems.

Polynomial Division

Polynomial division is a key algebraic tool which is similar to long division conducted with numbers. Here, you take a polynomial (dividend) and divide it by another polynomial (divisor), simplifying the expression across the division.
The process involves these critical steps:

• Identify the leading term in both the dividend and the divisor.
• Determine how many times the leading term of the divisor goes into the leading term of the dividend.
• Multiply this result by the entire divisor and subtract this from the original dividend.
• Repeat the process with the new dividend until the remainder is smaller than the leading term of the divisor.

In the context of our exercise, after applying polynomial division to the expression \(6x^2 - 7x - 5\) by \(3x - 5\), we get \(2x + 1\) as the quotient, which represents the height of the parallelogram.
This demonstrates how polynomial division helps simplify complex expressions into more digestible parts.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operational symbols. They are the fundamental building blocks of algebra and can represent various forms of relationships and equations.
Understanding algebraic expressions involves recognizing components like coefficients, variables, and constants. For example, in the expression \(6x^2 - 7x - 5\),

• \(6x^2\) represents a term with 6 as a coefficient and \(x^2\) as the variable part.
• \(-7x\) is another term with \(-7\) as the coefficient and \(x\) as the variable.
• \(-5\) is a constant term.

Manipulating these expressions requires skills such as factoring, expanding, and simplifying. These operations are essential in tasks like solving for the height of a parallelogram or any real-world problem involving equations.
By transforming complex expressions into simpler ones, using techniques like the polynomial division discussed earlier, you effectively apply algebraic skills to find solutions efficiently.