2/x+5 - x/x-5 =1
I multiplied both side by (x+5) (x-5) or by x^2 -25 and I arrive at
-x^3+x^2+25x-25 =0 And now I am stuck. Thanks for any help
When you multiplied, you went astray. You should have canceled common factors.
Rather than multiplying both sides by (x+5)(x-5), you should have just placed everything over the common denominator. That means you should have gotten
2/(x+5) - x/(x-5) =1
2(x-5) / (x+5)(x-5) - x(x+5) / (x+5)(x-5) = 1(x+5)(x-5) / (x+5)(x-5)
That is, each fraction is just multiplied by the missing factor of the LCD. Now the equation is just
2(x-5) - x(x+5) = (x-5)(x+5)
2x-10 - x^2-5x = x^2-25
2x^2+3x-15 = 0
Now just use the quadratic formula.
Thanks
To solve the equation 2/x+5 - x/x-5 = 1, you initially multiplied both sides of the equation by (x+5)(x-5) (or x^2 - 25) to eliminate the denominators. However, it seems like there was a mistake in the expansion of the expression. Let's rework the problem step by step:
Starting with the original equation: 2/(x + 5) - x/(x - 5) = 1
To eliminate the denominators, we need to find a common denominator for the two fractions on the left-hand side. The least common denominator here is (x + 5)(x - 5), as you correctly identified.
Next, multiply every term in the equation by (x + 5)(x - 5):
[(2/(x + 5)) * (x + 5)(x - 5)] - [(x/(x - 5)) * (x + 5)(x - 5)] = 1 * (x + 5)(x - 5)
Simplifying this expression:
[2(x - 5)] - [x(x + 5)] = (x + 5)(x - 5)
Expand and combine like terms:
2x - 10 - (x^2 + 5x) = x^2 + 5x - 25
Now, bring all the terms to one side of the equation:
2x - 10 - x^2 - 5x = x^2 + 5x - 25
Rearrange the terms:
-x^2 - 3x + 15 = 0
At this point, if you are stuck, you can utilize various methods to solve this quadratic equation, such as factoring, completing the square, or using the quadratic formula.
In this case, let's solve the equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation -x^2 - 3x + 15 = 0, a = -1, b = -3, and c = 15.
Substituting these values into the quadratic formula:
x = (-(-3) ± √((-3)^2 - 4(-1)(15))) / (2(-1))
x = (3 ± √(9 + 60)) / (-2)
x = (3 ± √69) / (-2)
Therefore, the solutions to the equation are given by:
x = (3 + √69) / -2
x = (3 - √69) / -2
This provides the two potential values for x that satisfy the original equation.
I hope this explanation helps you understand how to solve the equation step by step.