Verify if the equation has any solutions 2x/(x+5)-x/(x-5)=50/(25-x^2) .
2 Answers
justaguide | College Teacher | (Level 2) Distinguished Educator
We have to determine if 2x/(x+5) - x/(x-5) = 50/(25-x^2) has any solutions.
2x/(x+5) - x/(x-5) = 50/(25-x^2)
=> [2x(x - 5) - x(x + 5)] / (x^2 - 25) = 50 / (25-x^2)
=> 2x^2 - 10x - x^2 - 5x + 50 = 0
=> x^2 - 15x + 50 = 0
=> x^2 - 10x - 5x + 50 = 0
=> x(x - 10) - 5(x - 10) = 0
=> (x - 5)(x - 10) = 0
x = 5 and x = 10
But x = 5 makes x/(x-5) indeterminate
So the required solution of the equation is x = 10
giorgiana1976 | College Teacher | (Level 3) Valedictorian
First, we'll impose the constraints of existence of the fractions. All denominators have to be different from zero, for the fractions to exist.
x + 5 different from 0 => x different from -5
x - 5 different from 0 => x different from 5
The solutions of the equation can have any real value, except the values {-5 ; 5}.
We'll solve the equation:
2x(x-5) - x(x+5) = 50
2x^2 - 10x - x^2 - 5x = 50
We'll combine like terms:
x^2 - 15x - 50 = 0
We'll apply quadratic formula:
x1 = [15+sqrt(225 - 200)]/2
x1 = (15 + 5)/2
x1 = 10
x2 = (15-5)/2
x2 = 5
Since x2 = 5 is an exceted value, we'll reject this solution.
The equation will have only one real solution: x = 10.