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Chapter 1: Problem 128

Solve each equation by the method of your choice. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

Short Answer

Expert verified
The solutions are \(x = 1.5 + 0.5\sqrt{5}\) and \(x = 1.5 - 0.5\sqrt{5}\).

Step by step solution

01

Factor the denominator

Factor the denominator \(x^{2}-5x+6\) into \((x-2)(x-3)\). Now, you can see it becomes the common denominator for all fractions in the equations.
02

Multiply every term with common denominator

Multiply the entire equation by the common denominator \((x-2)(x-3)\). The equation would simplify to \[(x-1)(x-3) + x(x-2) = 1.\] Now we have a quadratic equation.
03

Simplify and rewrite the equation

Simplify the expression by expanding and combining like terms: \[x^2 - 4x + 3 + x^2 - 2x = 1\], this simplifies to: \(2x^2 - 6x + 3 = 1\).
04

Solve the quadratic equation

To solve the quadratic equation, subtract 1 from both sides to set the equation to zero: \(2x^2 - 6x + 2 = 0\). Divide all terms by 2 and the equation becomes \(x^2 - 3x + 1 = 0\). Find the roots by applying the quadratic formula: \(x = [-b \pm \sqrt{(b^2 - 4ac)}] / 2a\), where \(a=1\), \(b=-3\), and \(c=1\).

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Most popular questions from this chapter

Solve inequality using a graphing utility. Graph side separately. Then determine the values of \(x\) for which the graph for the left side lies above the graph for the right side. \(-3(x-6)>2 x-2\)

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Use interval notation to express solution sets and graph each solution set on a number line. Solve linear inequality. \(6(x-1)-(4-x) \geq 7 x-8\)

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Solve absolute value inequality. \(-4|1-x|<-16\)
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