/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 132 Solve each equation. $$\frac{x... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 132

Solve each equation. $$\frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6}$$

Short Answer

Expert verified
The solutions to the equation are \(x_1 = 1.75\) and \(x_2 = 0.25\).

Step by step solution

01

Rearrange the Equation

In order to approach the problem, it is best to move all terms to one side of the equation in order to simplify and solve later. Possible way to rearrange the equation would be to move the \(\frac{1}{x^{2}-5 x+6}\) to the left side, resulting in: \[\frac{x-1}{x-2}+\frac{x}{x-3}-\frac{1}{x^{2}-5 x+6}=0.\]
02

Find a Common Denominator

Since we have rational equations, we need to find a common denominator to combine the terms. In this case, \(x-2\), \(x-3\), and \(x^{2}-5 x+6\) are the denominators. The denominator \(x^{2}-5 x+6\) can be factored as \((x-2)(x-3)\). So, the common denominator will be \((x-2)(x-3)\). Rewrite all the fractions with this denominator.
03

Simplify the Equation

Multiplying the entirety of both sides of the equation by the common denominator \((x-2)(x-3)\) helps in clearing all fractions. With the fractions gone, you can simplify the equation, resulting in: \(x(x-3)+(x-1)(x-2)-1=0\).
04

Solve for x

The resultant equation is quadratic and can be solved in a way similar to simple quadratic equations. First simplify the equation and set it equal to zero: \[(x^{2}-3x)+(x^{2}-3x+2)-1=0\] Simplify and find roots: \(2x^{2}-6x+1=0\] Using the quadratic formula, \(x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) calculate x.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I subtracted \(\frac{3 x-5}{x-1}\) from \(\frac{x-3}{x-1}\) and obtained a constant.

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x,\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

$$\text { Solve for } t: \quad s=-16 t^{2}+v_{0} t$$

What is the discriminant and what information does it provide about a quadratic equation?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.