Problem 29 - Precalculus : Building Concepts and Connections

Question

Solve the rational equation. Check your solutions. $$\frac{1}{x^{2}-x-6}+\frac{3}{x+2}=\frac{-4}{x-3}$$

Step-by-Step Solution

Answer

This equation does not factor nicely, indicating that the solutions will not be rational numbers. Therefore, the solutions to this equation can only be found using numerical methods or by using the cubic equation solver on a scientific calculator or software like Wolfram Alpha.”

Step 1: Identify Common Denominator

Identify the Least Common Denominator (LCD) of the fractions. Here, LCD is the product of \(x^{2}-x-6\), \(x+2\) and \(x-3\).

Step 2: Multiply Each Term by the LCD

Multiply each term of the equation by the LCD to clear the fractions. \((x^{2}-x-6) \cdot (x+2) \cdot (x-3) \cdot (\frac{1}{{x^{2}-x-6}}) + (x^{2}-x-6) \cdot (x+2) \cdot (x-3) \cdot (\frac{3}{{x+2}})=(x^{2}-x-6) \cdot (x+2) \cdot (x-3) \cdot (\frac{-4}{{x-3}})\)

Step 3: Simplify the Equation

Simplify the equation by cancelling out the similar terms from the numerator and denominator. After simplifying, the equation will be \(x(x+2)(x-3) +3(x^{2}-x-6)(x-3) = -4(x^{2}-x-6)(x+2)\).

Step 4: Expand and Simplify

Expand the brackets and simplify the equation to get \(x^{3}-x^{2}-x=-4x^{2}+8x+24\).

Step 5: Set Equal to Zero

Move all the terms to one side of the equation to set it equal to zero. This results in \(x^{3}-3x^{2}+9x+24=0\).

Step 6: Solve for x

Now solve for x using synthetic division or by factoring. Unfortunately, this equation does not factor nicely, indicating that the solutions will not be rational numbers. Therefore, the solutions to this equation can only be found using numerical methods like Newton's method or by using the cubic equation solver on a scientific calculator or software like Wolfram Alpha.

Key Concepts

Least Common Denominator

In rational equations, we often deal with fractions that need a common foundation before being solved. The Least Common Denominator (LCD) is that shared foundation. It's essential for simplifying and solving equations involving any fractions. The LCD for a set of fractions is the smallest expression that each of the denominators can divide without leaving a remainder.
To find the LCD, identify all the distinct factors in the denominators of your fractions. Multiply these factors to form the LCD. For example, if you have denominators like \(x^2-x-6\), \(x+2\), and \(x-3\), the LCD would be the product of these three expressions: \((x^{2}-x-6)(x+2)(x-3)\).
Once you have the LCD, multiply each term in the equation by this common denominator to eliminate the fractions. This brings the equation to a form that is much simpler to work with, helping to solve it efficiently.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, especially when dividing by a binomial of the form \(x-c\). This technique is favored because it's less cumbersome than long division and quicker to execute.
To apply synthetic division to a polynomial like \(x^3-3x^2+9x+24\), we first choose a value for \(c\). We arrange the polynomial's coefficients in a row and begin by bringing down the leading coefficient to the row directly beneath. Next, multiply this coefficient by \(c\), placing the result under the next coefficient. Add these to get the new number below it, which you'll multiply again by \(c\). Continue this process through the coefficients, yielding both a quotient and a remainder depending on the degree of the original polynomial.
Synthetic division simplifies solving polynomial equations, especially during intermediate steps of rational equations where the traditional methods are less efficient.

Cubic Equation Solver

Cubic equations can be more intricate to solve because they take the form of \(ax^3+bx^2+cx+d=0\), where \(aeq0\). In many instances, these equations do not factor neatly into rational numbers, making manual solutions impractical. Here, cubic equation solvers or numerical methods become invaluable tools.
One method is to use software or calculators that employ algorithms rooted in numerical analysis to supply accurate approximations of solutions. Popular options include Wolfram Alpha, which is accessible online for solving equations beyond simple factoring.
These solvers work by applying formulas or iterative methods like the Newton-Raphson method to find precise roots of the equation, thus solving it efficiently when algebraic methods might fail or be too complex. Cubic equation solvers help bridge the gap when equations do not easily reveal solutions, allowing you to find them with precision and ease.