hi jiskha. how do you solve x-4/x+2 + 2/x-2 = 17/x^2-4? I factored it so it became x-4(x+2)(x-2)+2(x+2)(x-2)=17, but I don't know where to go from there. thanx!!
not quite,
multiply each term by x^2-4, realizing that x^2-4 = (x+2)(x-2)
so you would get
(x-4)(x-2) + 2(x+2) = 17
x^2 - 6x + 8 + 2x + 4 = 17
x^2 -4x - 5 = 0
(x-5)(x+1) = 0
x = 5 or x = -1
To solve the equation (x - 4)/(x + 2) + 2/(x - 2) = 17/(x^2 - 4), you are on the right track by factoring the denominators and combining the terms.
Here's the next step:
First, distribute the factors in the numerators:
(x - 4)(x + 2)(x - 2) + 2(x + 2)(x - 2) = 17
Now, expand each term:
(x^2 - 4)(x - 2) + 2(x + 2)(x - 2) = 17
Multiply each term by the common denominator, (x^2 - 4):
(x^2 - 4)(x - 2) + 2(x + 2)(x - 2) = 17(x^2 - 4)
Now, simplify and combine like terms:
(x^3 - 2x^2 - 4x + 8) + 2(x^2 - 4) = 17x^2 - 68
Expand and simplify further:
x^3 - 2x^2 - 4x + 8 + 2x^2 - 8 = 17x^2 - 68
Next, combine like terms:
x^3 + x^2 - 4x = 17x^2 - 68
Rearrange the equation by moving all the terms to one side:
x^3 + x^2 - 17x^2 - 4x + 68 = 0
Simplify further:
x^3 - 16x^2 - 4x + 68 = 0
At this point, you have a cubic equation that may require factoring, using the rational root theorem, or using numerical methods such as graphing or a calculator to find the solutions. Unfortunately, there is no simple algebraic method to solve a general cubic equation like this.
You can use a calculator or computer software to find the approximate solutions or apply numerical methods to find more accurate solutions if needed.