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Chapter 1: Problem 78

Find all real solutions of the equation. $$\frac{1}{x-1}-\frac{2}{x^{2}}=0$$

Short Answer

Expert verified
There are no real solutions to the equation.

Step by step solution

01

Identify Common Denominator

To solve the equation \(\frac{1}{x-1}-\frac{2}{x^{2}}=0\), we first look for a common denominator to combine the fractions. The common denominator for \(x-1\) and \(x^2\) is \(x^2(x-1)\).
02

Rewrite Each Term

Rewrite each term with the common denominator: \(\frac{x^2}{x^2(x-1)} - \frac{2(x-1)}{x^2(x-1)} = 0\), which simplifies to \(\frac{x^2 - 2(x-1)}{x^2(x-1)} = 0\).
03

Simplify and Set Numerator to Zero

Simplify the expression: \(x^2 - 2x + 2 = 0\) and set the numerator to zero, as the fraction equals zero only when its numerator is zero: \(x^2 - 2x + 2 = 0\).
04

Solve the Quadratic Equation

Attempt to solve the quadratic equation \(x^2 - 2x + 2 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1, b=-2, c=2\).
05

Discriminant Calculation

Calculate the discriminant \(b^2 - 4ac = (-2)^2 - 4(1)(2) = 4 - 8 = -4\). The discriminant is negative, indicating there are no real solutions.

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Key Concepts

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

Common Denominator
When dealing with rational equations like \( \frac{1}{x-1} - \frac{2}{x^2} = 0 \), finding a common denominator is crucial. This process allows us to combine fractions into a single term, making it easier to solve the equation.
To find the common denominator, identify the distinct factors in each denominator. In our problem, the fractions have denominators \( x-1 \) and \( x^2 \). The common denominator must include both factors: \( x^2(x-1) \).
  • Multiply each term by this common denominator.
  • Rewrite the equation: \( \frac{x^2}{x^2(x-1)} - \frac{2(x-1)}{x^2(x-1)} = 0 \).
This allows the subtraction of fractions and simplifies the equation to focus only on the numerators.
Quadratic Formula
Once the rational equation is simplified, you might find yourself with a quadratic equation like \( x^2 - 2x + 2 = 0 \). Solving this requires the quadratic formula, a cornerstone in algebra for finding the roots of any quadratic equation.
The quadratic formula is written as:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a \), \( b \), and \( c \) are coefficients from the standard quadratic equation form \( ax^2 + bx + c = 0 \).
Using these coefficients:
  • \( a = 1 \)
  • \( b = -2 \)
  • \( c = 2 \)
Simply plug them into the formula to find the values of \( x \). Understanding this powerful formula helps solve various quadratic equations, even if they initially seem unsolvable.
Discriminant
The discriminant is a key part of the quadratic formula. It helps determine the nature of the roots, and is represented by \( b^2 - 4ac \). In our case for \( x^2 - 2x + 2 = 0 \), calculate it as:
\[ (-2)^2 - 4 \times 1 \times 2 = 4 - 8 = -4 \]
A negative discriminant tells us there are no real roots. Why? Because the square root of a negative number is imaginary, indicating complex solutions.
  • If \( b^2 - 4ac > 0 \): Two real and distinct roots.
  • If \( b^2 - 4ac = 0 \): One real root, repeated.
  • If \( b^2 - 4ac < 0 \): No real roots, but two complex roots.
Knowing how to calculate and interpret the discriminant quickly answers whether real solutions exist.

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