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Simplifying this further gives us the solution x = −3. It is easily checked that none of the zeros of x ( x + 1)( x + 2) – namely x = 0 , x = −1 , and x = −2 – is a solution of the final equation, so no spurious solutions were introduced.
The solution = is in fact a valid solution to the original equation; but the other solution, =, has disappeared. The problem is that we divided both sides by x {\displaystyle x} , which involves the indeterminate operation of dividing by zero when x = 0. {\displaystyle x=0.}
In the second step, they were divided by 3. The final result, 4 / 3 , is an irreducible fraction because 4 and 3 have no common factors other than 1. The original fraction could have also been reduced in a single step by using the greatest common divisor of 90 and 120, which is 30. As 120 ÷ 30 = 4, and 90 ÷ 30 = 3, one gets
In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1. [14] [15] It is said to be an improper fraction, or sometimes top-heavy fraction, [16] if the absolute value of the fraction is greater than or equal to 1 ...
Continued fractions can also be applied to problems in number theory, and are especially useful in the study of Diophantine equations. In the late eighteenth century Lagrange used continued fractions to construct the general solution of Pell's equation, thus answering a question that had fascinated mathematicians for more than a thousand years. [9]
This equation can be viewed as a Diophantine equation, that is, an equation for which only integer solutions are sought. In this case, the solution set is the empty set, since 2 is not the square of an integer. However, if one searches for real solutions, there are two solutions, √ 2 and – √ 2; in other words, the solution set is {√ 2 ...
For example, a fraction is put in lowest terms by cancelling out the common factors of the numerator and the denominator. [2] As another example, if a × b = a × c , then the multiplicative term a can be canceled out if a ≠0, resulting in the equivalent expression b = c ; this is equivalent to dividing through by a .
Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. [4] Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction ...