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Chapter 7: Problem 30

\(\frac{6 x+12 y}{x^{2}-x y-6 y^{2}}\)

Short Answer

Expert verified
The simplified form of the fraction \(\frac{6 x+12 y}{x^{2}-x y-6 y^{2}}\) is \(\frac{6(x+2y)}{(x-2y)(x+3y)}\)

Step by step solution

01

Factor the Numerator

In the expression \(\frac{6 x+12 y}{x^{2}-x y-6 y^{2}}\), the common factor in the numerator is 6. Factoring it out, we get: \(6(x+2y)\)
02

Factor the Denominator

Next, we factor the denominator. The equation \(x^{2}-x y-6 y^{2}\) is a quadratic equation, which can be factored into two binomials: \((x-2y)(x+3y)\)
03

Simplify the Fraction

Finally, substitute the factored numerator and denominator back into the fraction. We find no common factors between the numerator and the denominator that can be cancelled out. Therefore, the simplified form of the original fraction is \(\frac{6(x+2y)}{(x-2y)(x+3y)}\)

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Key Concepts

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

Understanding Numerators and Denominators
In any fraction, the number above the line is called the numerator, and the number below the line is the denominator. The fraction presents a division of the numerator by the denominator. For example, in the expression \( \frac{6x + 12y}{x^2 - xy - 6y^2} \), \(6x + 12y\) is the numerator and \(x^2 - xy - 6y^2\) is the denominator.

The numerator tells you how many parts you have, while the denominator tells you how many equal parts the whole is divided into.
  • If the numerator is greater than the denominator, your fraction is greater than 1.
  • If the numerator is smaller, the fraction is less than 1.
  • When they are equal, the fraction equals 1.
Recognizing the roles of the numerator and denominator is essential when manipulating expressions, especially for factoring as it details how the terms share common factors.
The Basics of Quadratic Equations
Quadratic equations are mathematical statements where the highest exponent of the variable is 2. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). In our context, \(x^2 - xy - 6y^2\) represents a quadratic expression, though not set to zero.

Factoring quadratic equations involves expressing them as a product of two binomials. Here, the expression factors into \((x - 2y) (x + 3y)\). To factor:
  • Look for two numbers that multiply to give ac (the product of a and c in \(ax^2 + bx + c\)) and add to give b.
  • Rewriting the expression using these numbers and factoring by grouping can help reveal the pairs.
Working with quadratic equations allows for simplification and finding solutions or roots when set to zero.
Simplifying Fractions in Algebra
Simplifying fractions involves reducing them to their simplest form, where no common factors exist between the numerator and the denominator, aside from 1. In algebra, this not only involves numbers but also variables within expressions. Let's look at \( \frac{6(x + 2y)}{(x - 2y)(x + 3y)} \).

To simplify:
  • First, factor both the numerator and the denominator.
  • Cancel out any common factors that appear in both the numerator and denominator.
In this case, after factoring, we find nothing to cancel out further. This makes \(\frac{6(x + 2y)}{(x - 2y)(x + 3y)}\) the simplest form. Simplification is crucial as it prepares expressions for further calculations and helps uncover relationships between variables.

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Most popular questions from this chapter

\(4 x^{2}-x^{3}-8+2 x\)

\(x^{3}-x^{2}+4 x-4\)

\(\frac{5}{2}-\frac{12}{x-4}=6\)

Work Rate One person takes \(1 \frac{1}{2}\) times as long to rake a front lawn as a second person. The two people can rake the front lawn together in 30 minutes. Find the time it takes the first person to rake the front lawn.

\(\frac{15}{x}-4=\frac{6}{x}+3\)
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