Solve for x.
x/(x-2) + (2x)/[4-(x^2)] = 5/(x+2)
Please show work!
putting everything over a common denominator of (x-2)(x+2), we have
x^2 = 5(x-2)
x^2 - 5x + 10 = 0
then just solve that for x
Make sure that x ≠ ±2 since those values are not allowed in the original equation
x/(x-2) + (2x)/[4-(x^2)] = 5/(x+2)
This calls for finding the common denominator, and then solving for x using the numerators.
Do not forget to exclude the asymptotes values of x where the denominator becomes zero.
LCM (lowest ommon multiple) of denominators (x-2), (4-x²) and (x+2) is (x²-4), since (x-2)(x+2)=(4-x²)
Provided that x≠2 and x≠-2, then we can write the equation above as:
[x(x+2)-2x]/(x²-4) = 5(x-2)/(x²-4)
provided x≠2 and x≠-2
Equating numerators,
x²+2x-2x=5(x-2)
=>
x²-5x+10=0
However, the solution does not have real roots. The complex roots are:
x=(5±√*i)/2
x=(5±(√15)*i)/2
If you do not expect complex roots, please check for typos in the original question.
Thank you guys! That is what I got. I wasn't expecting the complex roots.. That's why I wanted to check whether or not I was doing it right!
You're welcome!
To solve for x in the given equation, we can follow these steps:
Step 1: Simplify the equation by finding the least common denominator (LCD) for the denominators on both sides of the equation.
The denominators on the left side are (x - 2) and (4 - x^2). To find the LCD, we need to factor the second denominator:
4 - x^2 = (2 - x)(2 + x)
The LCD is then (x - 2)(2 - x)(2 + x).
Step 2: Multiply each term by the LCD to eliminate the fractions.
(x - 2)(2 - x)(2 + x) [x/(x - 2) + (2x)/(4 - x^2)] = (x - 2)(2 - x)(2 + x)(5/(x + 2))
Simplifying the left side:
(x - 2) cancels out with (x - 2) in the first term, and (2 - x) in the second term cancels out with (2 + x) in the third term:
(x + 2x)(2 + x) = 5(x - 2)
3x(2 + x) = 5(x - 2)
Step 3: Expand and simplify.
6x + 3x^2 = 5x - 10
Rearranging the terms:
3x^2 - x - 10 = 0
Step 4: Solve the quadratic equation.
To solve the quadratic equation, we can factor or use the quadratic formula.
Factoring the quadratic equation may not be easy in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For the quadratic equation 3x^2 - x - 10 = 0, the values of a, b, and c are:
a = 3, b = -1, c = -10
Plugging these values into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(3)(-10))) / (2(3))
Simplifying:
x = (1 ± √(1 + 120)) / 6
x = (1 ± √121) / 6
x = (1 ± 11) / 6
So, the solutions for x are:
x = (1 + 11) / 6 = 12/6 = 2
and
x = (1 - 11) / 6 = -10/6 = -5/3