Solve the rational equation.
3/(x)+6/(x+3)=8/(x+1)
To solve the rational equation, we can start by finding a common denominator for the fractions on both sides of the equation. In this case, the common denominator is (x)(x+3)(x+1).
Now, we can multiply every term of the equation by the common denominator to eliminate the denominators:
[(x)(x+3)(x+1)] * [3/(x)] + [(x)(x+3)(x+1)] * [6/(x+3)] = [(x)(x+3)(x+1)] * [8/(x+1)]
After multiplying, we get:
3(x+3)(x+1) + 6(x)(x+1) = 8(x)(x+3)
Now, we can simplify the equation by expanding and combining like terms:
3(x^2+4x+3) + 6x(x+1) = 8x(x+3)
3x^2 + 12x + 9 + 6x^2 + 6x = 8x^2 + 24x
Now, let's move all terms to one side of the equation:
9 + 24x - 18x - 8x^2 - 9x^2 - 12x = 0
-17x^2 - 6x + 9 = 0
This quadratic equation can be solved by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the solutions for x:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = -17, b = -6, and c = 9. Plugging in these values, we get:
x = (-(-6) ± √((-6)^2 - 4(-17)(9))) / (2(-17))
x = (6 ± √(36 + 612)) / (-34)
x = (6 ± √648) / (-34)
x = (6 ± 24) / (-34)
x = (6 + 24) / (-34) or x = (6 - 24) / (-34)
x = 30 / (-34) or x = -18 / (-34)
Simplifying these fractions, we get:
x = -15/17 or x = 9/17
Therefore, the solutions to the rational equation 3/(x) + 6/(x+3) = 8/(x+1) are x = -15/17 and x = 9/17.