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Chapter 0: Problem 131

Solve each equation. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

Short Answer

Expert verified
The solution to the equation \(\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}\) is \(x=1\).

Step by step solution

01

Start simplifying

Begin by rewriting the given equation: \(\frac{1}{x^{2}-3x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}\) as \(\frac{1}{(x-1)(x-2)}=\frac{1}{x+2}+\frac{5}{(x+2)(x-2)}\) this is done to factorise the denominators.
02

Find a common denominator

The next step involves summing up on the righthand side of the equation by using a common denominator, \((x+2)(x-2) = x^{2}-4\). The equational form then is \(\frac{1}{(x-1)(x-2)}=\frac{(x^2-4)+5}{x^{2}-4}\) which transforms into \(\frac{1}{(x-1)(x-2)}=\frac{x^2+1}{x^{2}-4}\).
03

Simplification and formation of a quadratic equation

The equation can then be simplified further by multiplying the entire equation by \((x-1)(x-2)(x^{2}-4)\) to rid the equation of fractions. This gives a quadratic equation: \((x^{2}-4)(x-1) = (x^{2}+1)(x-2)\).
04

Expand and simplify

Expand and simplify the equation to bring it into standard form: \(x^3-5x = x^3 -x^2 -2x +2\). Then move every term to one side of the equation, which simplifies to: \(x^2 -3x +2 = 0\).
05

Solve for \(x\)

Apply the root formula to get solutions. The roots of the equation are \(x=1\) and \(x=2\).
06

Check roots

The roots should be checked if they satisfy the original equation or not. The denominator in the given equation should not be zero. The denominators \(x^2-3x+2\), \(x+2\) and \(x^2-4\) give the sets of roots \({1, 2}\), \(-2\) and \({2, -2}\) respectively. Here, the root \(x=2\) causes one or more of the denominators to be zero and is therefore not a valid solution. Hence, the only valid solution is \(x=1\).

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. Parts for an automobile repair cost \(\$ 175 .\) The mechanic charges \(\$ 34\) per hour. If you receive an estimate for at least \(\$ 226\) and at most \(\$ 294\) for fixing the car. what is the time interval that the mechanic will be working on the job?

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Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 2800 pounds. If the elevator operator weighs 265 pounds and each cement bag weighs 65 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

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