okay for the first one i/ve done it twice and both times i got 10, not -10. check your work to make sure you did it correctly.
for the second one i got 24 so you are correct
Directions solve for x.
(x)/(x-2)-(x+1)/(x)=(8)/(x^2-2x)
The answer i get is :
-10=x or x=-10
Problem #2
solve for x
(x)/(6)-(x)/(8)=1
The answer i get is: x = 24
To solve the first equation, we need to simplify the left side by finding a common denominator for the two fractions. The common denominator for (x)/(x-2) and (x+1)/(x) is x(x-2).
So, let's rewrite the equation with the common denominator:
(x(x))/(x(x-2)) - ((x+1)(x-2))/(x(x-2)) = (8)/(x^2 - 2x)
Now we can combine the fractions on the left side:
(x^2)/(x(x-2)) - (x^2 - x - 2)/(x(x-2)) = (8)/(x^2 - 2x)
Next, we can combine the numerators:
(x^2 - (x^2 - x - 2))/(x(x-2)) = (8)/(x^2 - 2x)
Simplifying further:
(x^2 - x^2 + x + 2)/(x(x-2)) = (8)/(x^2 - 2x)
Simplifying the numerator and denominator:
(x + 2)/(x(x-2)) = (8)/(x^2 - 2x)
Now we can cross-multiply:
(x + 2)(x^2 - 2x) = 8(x(x-2))
Expanding the multiplication on both sides:
x^3 - 2x^2 + 2x^2 - 4x = 8x^2 - 16x
Simplifying:
x^3 - 4x = 8x^2 - 16x
Rearranging the equation:
x^3 - 16x^2 = 12x
Factoring out an x:
x(x^2 - 16x) = 12x
Dividing both sides by x:
x^2 - 16x = 12
Bringing all terms to one side:
x^2 - 16x - 12 = 0
Now we can factor or use the quadratic formula to solve for x. I'll use the quadratic formula here:
x = (-(-16) ± sqrt((-16)^2 - 4(1)(-12))) / (2(1))
Simplifying inside the square root:
x = (16 ± sqrt(256 + 48)) / 2
x = (16 ± sqrt(304)) / 2
x = (16 ± 2sqrt(76)) / 2
Simplifying further:
x = 8 ± sqrt(76)
So, the possible values for x are:
x = 8 + sqrt(76)
x = 8 - sqrt(76)
Now let's check if x = -10 is a valid solution. Substituting -10 into the original equation:
(-10)/(-10-2)-(-10+1)/(-10)=(8)/((-10)^2 - 2(-10))
Simplifying:
(-10)/(-12)-(-9)/(-10)=(8)/(100 + 20)
(10/12)-(9/10)=(8)/(120+20)
Multiplying both sides by 120 (to clear the denominators):
1000 - 1080 = 960
This is not true, so x = -10 is not a valid solution. Therefore, we only have solutions for:
x = 8 + sqrt(76)
x = 8 - sqrt(76)
Now, let's move on to the second equation.
To solve the equation (x)/(6) - (x)/(8) = 1, we need to find a common denominator for (x)/(6) and (x)/(8). The least common multiple of 6 and 8 is 24.
So, let's rewrite the equation with the common denominator:
(8x)/(24) - (6x)/(24) = 1
Now, we can combine the fractions on the left side:
(8x - 6x)/(24) = 1
Simplifying:
2x/24 = 1
Next, we can simplify further by dividing both sides by 2:
x/12 = 1
Multiply both sides by 12 to isolate x:
x = 12
So, the solution for the second equation is x = 12.
Now, let's verify this solution by substituting x = 12 into the original equation:
(12)/(6) - (12)/(8) = 1
Simplifying:
2 - 1.5 = 1
0.5 = 1
This is not true, so x = 12 is not a valid solution for the original equation.
In conclusion, the corrected solution for the first equation is:
x = 8 + sqrt(76)
x = 8 - sqrt(76)
And the solution for the second equation is:
x = 12