Question

розв'яжіть рівняння -х(х-2)(2-х)=(х+4)(х-1)-х
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Answer 1

Ответ:

To solve the equation \(-x(x-2)(2-x) = (x+4)(x-1)-x\), let's simplify both sides:

Expanding the products and combining like terms:

\[-x(2x^2 - 4x) = (x^2 + 3x - 4) - x\]

Distribute the -x on the left side:

\[-2x^3 + 4x^2 = x^2 + 2x - 4 - x\]

Combine like terms on both sides:

\[-2x^3 + 4x^2 = x^2 + x - 4\]

Move all terms to one side of the equation:

\[-2x^3 + 3x^2 - x + 4 = 0\]

Now, try to factor the cubic equation. A potential factor is \(x - 1\):

\[(x - 1)(-2x^2 + x - 4) = 0\]

The quadratic factor doesn't factor easily, so the solutions are \(x = 1\) (from \(x - 1 = 0\)) and the roots of the quadratic factor can be found using the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

For \(-2x^2 + x - 4\), where \(a = -2\), \(b = 1\), and \(c = -4\). The solutions are complex numbers.

So, the real solution is