Let's first simplify the equation on both sides.
-2x(1-x) + 3(x-4) = -x(x+2) - 6x
Expanding the terms gives:
-2x + 2x^2 + 3x - 12 = -x^2 - 2x - 6x
Combining like terms:
2x^2 - 2x + 3x - 12 = -x^2 - 2x - 6x
This simplifies to:
2x^2 + x - 12 = -x^2 - 8x
Now, let's combine like terms and move all terms to one side of the equation:
2x^2 + x - 12 + x^2 + 8x = 0
3x^2 + 9x - 12 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 3, b = 9, and c = -12. Plugging into the formula:
x = (-9 ± sqrt(81 + 144)) / x = (-9 ± sqrt(225)) / x = (-9 ± 15) / 6
This gives two solutions:
x = (6/6) = x = (-24/6) = -4
Therefore, the solutions to the equation are x = 1 and x = -4.
Let's first simplify the equation on both sides.
-2x(1-x) + 3(x-4) = -x(x+2) - 6x
Expanding the terms gives:
-2x + 2x^2 + 3x - 12 = -x^2 - 2x - 6x
Combining like terms:
2x^2 - 2x + 3x - 12 = -x^2 - 2x - 6x
This simplifies to:
2x^2 + x - 12 = -x^2 - 8x
Now, let's combine like terms and move all terms to one side of the equation:
2x^2 + x - 12 + x^2 + 8x = 0
3x^2 + 9x - 12 = 0
This is a quadratic equation, which can be solved using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 3, b = 9, and c = -12. Plugging into the formula:
x = (-9 ± sqrt(81 + 144)) /
x = (-9 ± sqrt(225)) /
x = (-9 ± 15) / 6
This gives two solutions:
x = (6/6) =
x = (-24/6) = -4
Therefore, the solutions to the equation are x = 1 and x = -4.