Find all solutions (answers should be given in simplified form). Some answers may involve the number i.
(x^2/6) - (2x/3) = 3 + (x/2)
I turned (2x/3) into (4x/6) and subtracted it from (x^2/6) to get: x(x-4) / 6
I turned 3 into 6/2 and added it to (x/2) to get: (x+6) / 2
So that leaves me with:
x(x-4) / 6 = (x+6)/6
Where do I go from here?
Sorry, the final line should read:
x(x-4) / 6 = (x+6)/2
I think what "you do from here" is to expand and factor the terms; however, can you subtract 2x/3 from X^2/6. Those are not like terms. I don't believe you can do that, at least not legally. Therefore, I don't agree with the "getting here" part.
Consider using the original equation, multiplying everything by 6 (to clear the denominators), factor the terms and sove the equation. See if that doesn't give you x = -2 and x = +9. Then substitute those numbers into the original equation and see if the equation is satisfied.
To solve the equation x(x-4)/6 = (x+6)/2, you can eliminate the denominators by multiplying both sides of the equation by the least common multiple (LCM) of the two denominators.
The LCM of 6 and 2 is 6, so multiplying both sides of the equation by 6 gives:
6 * (x(x-4)/6) = 6 * ((x+6)/2)
Simplifying, you get:
x(x-4) = 3(x+6)
Expand the left side of the equation:
x^2 - 4x = 3x + 18
Reorder the terms:
x^2 - 4x - 3x - 18 = 0
Combine like terms:
x^2 - 7x - 18 = 0
Now, you need to factor the quadratic equation or use the quadratic formula to solve for x. In this case, the equation factors into:
(x + 2)(x - 9) = 0
Setting each factor equal to zero gives:
x + 2 = 0 or x - 9 = 0
Solving for x in each equation:
x = -2 or x = 9
So the solutions to the equation are x = -2 and x = 9.