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Chapter 1: Problem 52

Solve the equation. $$ \frac{m}{m-2}-\frac{27}{7}=\frac{2}{m^{2}-m-2} $$

Short Answer

Expert verified
The given equation is: \[ \frac{m}{m-2}-\frac{27}{7}=\frac{2}{m^{2}-m-2} \] Following the steps of finding the LCD, multiplying both sides by it, and simplifying, we get: \[ -10m^3 + 20m + 7 = 0 \] There is no simple algebraic method to solve this cubic equation. Using a graphical approach or a numerical method, we find the approximate real solutions for m: \(m \approx 0.453, m \approx -0.432\) (rounded to three decimal places)

Step by step solution

01

Find the least common denominator (LCD)

To find the LCD, we have to consider the denominators: (m-2), 7, and \((m^2- m- 2)\). We see that these denominators have no common factors. Therefore, the LCD will be the product of these denominators: \((m-2) \cdot 7 \cdot (m^2-m-2)\).
02

Multiply both sides of the equation by the LCD

Now, we will multiply each term on both sides of the equation by the LCD: \((m-2) \cdot 7 \cdot (m^2-m-2) \cdot \left(\frac{m}{m-2} - \frac{27}{7} - \frac{2}{m^2 - m - 2}\right)\)
03

Cancel out the denominators and simplify

When we multiply each term by the LCD, we will cancel out the denominators: \(7(m^2 - m - 2)\cdot m - (m-2)(m^2-m-2) \cdot 27 - 2(m-2) \cdot 7 = 0\) Now, we will distribute and simplify: \(7m^3 - 7m^2 - 14m - 27m^3 + 27m^2 + 54m - 14m + 28 = 0\) Combine like terms: \((-20m^3) + 40m + 14 = 0\)
04

Solve for m

To find the values of m, we can first divide the whole equation by 2 to make it simpler: \(-10m^3 + 20m + 7 = 0\) Now, we need to solve for m. Unfortunately, there is no simple algebraic method to solve this cubic equation. To express this in a general way, we need to follow complex number theorem. But, for real-values, we need to use a numerical method or graphical approach to find out the possible values of m. Generally, polynomial equations of higher degrees like this cannot be solved algebraically. However, we can use a graphing calculator, or online tool like Desmos or Wolfram Alpha to get the approximate real solutions for m: m 鈮� 0.453, m 鈮� -0.432 (rounded to three decimal places) Please note that these are approximate real values which may vary slightly depending on the method or calculator used.

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Key Concepts

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

Least Common Denominator
Understanding the least common denominator (LCD) is essential when working with fractions in algebra. The LCD of two or more fractions is the smallest number that all denominators can divide into without leaving a remainder.

Finding the LCD allows us to combine fractions by giving them a common denominator, simplifying the process of addition, subtraction, or comparison.

Practical Use of LCD in Solving Equations

When solving fractional equations, like in our exercise, multiplying each term by the LCD clears the fractions, making it easier to deal with polynomials or linear expressions. This is a key step in solving for variables that appear in multiple fractions with different denominators.
Polynomial Equations
Polynomial equations are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients.

They can range from simple linear equations (1st degree) to complex higher-degree equations. Cubic equations, or 3rd-degree polynomial equations, have the general form of \(ax^3 + bx^2 + cx + d = 0\), where \(a, b, c,\) and \(d\) are constants, and \(a \eq 0\).

Challenges in Solving Higher-Degree Polynomials

While quadratic equations (2nd degree) can be solved using the quadratic formula, higher-degree polynomial equations do not have a straightforward algebraic solution and often require numerical or graphical methods for their solutions.
Graphical Approach in Mathematics
The graphical approach in mathematics provides a visual method to understand and solve equations, including polynomials.

By plotting the function on a coordinate plane, the points at which the graph intersects the x-axis represent the solutions to the equation.

Benefits of Graphical Solutions

This method not only gives a tangible representation of the roots but also helps in understanding the behavior of the function, such as intervals of increase and decrease, local maxima and minima, and points of inflection. Despite its advantages, a graphical solution may not always provide the exact numerical value, especially for irrational or complex roots.
Numerical Methods in Algebra
Numerical methods in algebra involve algorithms and approximation techniques to find solutions to equations that are difficult or impossible to solve analytically.

These methods are particularly useful in solving high-degree polynomial equations, where traditional algebraic methods fall short.

Common Numerical Techniques

Techniques such as the Newton-Raphson method, bisection method, or synthetic division can iterate towards an approximate solution. In the context of the given exercise, numerical methods help to approximate the values of \(m\) where an algebraic solution to the cubic equation is not feasible. The use of a graphing tool or calculator employs these numerical algorithms behind the scenes to provide the approximate roots.

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