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Chapter 1: Problem 127

Solve each equation by the method of your choice. $$\frac{1}{x^{2}-3 x+2}=\frac{1}{x+2}+\frac{5}{x^{2}-4}$$

Short Answer

Expert verified
The solution to the equation is \(x = 2, 5, -0.5\)

Step by step solution

01

Simplify fractions

Simplify each fraction by factoring out if possible, \(x^{2}-3 x+2=x^{2}-2x-x+2=(x-1)(x-2)\) and \(x^{2}-4 = (x-2)(x+2)\). The equation can be rewritten as: \(\frac{1}{(x-1)(x-2)} = \frac{1}{x+2} + \frac{5}{(x-2)(x+2)}\)
02

Find Common Denominator

As the denominators are not the same, find the least common denominator (LCD) which is \( (x-1)(x-2)(x+2)\). Multiply through by the LCD: \((x+2)(x-2)+(x-1)(x+2)*5 =(x-1)(x+2)(x-2)\)
03

Expand and Simplify

Expand the brackets on both sides of the equation: \(x^2-4 + 5x^2 + 5x -10 = x^3 - x^2 -4x +4 \). Collect like terms and simplify: \(6x^2 +5x - 14 = x^3 - x^2 -4x + 4\). Put the equation in the standard form \(x^3 -7x^2 -9x + 10 = 0\)
04

Solve the equation

Find the solution using graphing, factoring, or the rational root theorem. For this specific case, x equals to 2, 5, and minus half (-0.5)

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Most popular questions from this chapter

Solve absolute value inequality. \(|x|<3\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Solve without squaring both sides: $$\text { Solve for } x: x^{6}+x^{3}-2 x^{2}-0$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \% .88 \% .92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of \(\$ 15\) with a charge of \(\$ 0.08\) per text. Plan \(\mathrm{B}\) has a monthly fee of \(\$ 3\) with a charge of \(\$ 0.12\) per text. How many text messages in a month make plan A the better deal?

Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(8 x+3>3(2 x+1)+x+5\)
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