Show that the equation (1) divided by (x+1) - (x)divided by (x-2)=0 has no real roots
Well, to begin, start with 1/(x+1-x). The x's cancel out because they are opposite signs, so now you have 1/1, or just 1. Then, you are dividing 1 by (x-2). In order for an expression to have roots, you have to be able to plug in the roots for x and get 0 out of the equation. In this equation, you might think that 2 is a root. However, that would make the denominator of the fraction 0, which makes the equation undefined. The only way for an equation that's a fraction to have roots is if the numerator = 0. There is no way to get that if the numerator is 1.
To show that the equation (1) divided by (x+1) - (x) divided by (x-2) = 0 has no real roots, we need to analyze the expression and find for which values of x it is equal to 0.
First, let's simplify the expression:
(1/(x+1)) - (x/(x-2)) = 0
Next, let's find a common denominator to combine the fractions:
[(x-2)/(x-2)] * (1/(x+1)) - [(x+1)/(x+1)] * (x/(x-2)) = 0
Simplifying further:
(x-2) / (x^2 - x - 2) - (x(x+1)) / (x^2 - x - 2) = 0
Combining the fractions:
((x-2) - x(x+1)) / (x^2 - x - 2) = 0
Expanding the numerator:
(x-2 - x^2 - x) / (x^2 - x - 2) = 0
Rearranging terms:
(-x^2 - 2x - 2) / (x^2 - x - 2) = 0
Now, in order for the expression to equal 0, the numerator must be equal to 0 while the denominator should not be equal to 0.
Setting the numerator equal to 0:
-x^2 - 2x - 2 = 0
To find the roots of this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -1, b = -2, and c = -2.
Plugging these values into the formula:
x = (-(-2) ± √((-2)^2 - 4(-1)(-2))) / (2(-1))
x = (2 ± √(4 - 16)) / (-2)
x = (2 ± √(-12)) / (-2)
Here, we encounter a problem. The √(-12) results in an imaginary number because there is no real number that squares to a negative value. Therefore, the expression has no real roots.
In conclusion, the equation (1) divided by (x+1) - (x) divided by (x-2) = 0 has no real roots.