-x^2+4x=x-20

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Solution for -x^2+4x=x-20 equation:


Simplifying
-1x2 + 4x = x + -20

Reorder the terms:
4x + -1x2 = x + -20

Reorder the terms:
4x + -1x2 = -20 + x

Solving
4x + -1x2 = -20 + x

Solving for variable 'x'.

Reorder the terms:
20 + 4x + -1x + -1x2 = -20 + x + 20 + -1x

Combine like terms: 4x + -1x = 3x
20 + 3x + -1x2 = -20 + x + 20 + -1x

Reorder the terms:
20 + 3x + -1x2 = -20 + 20 + x + -1x

Combine like terms: -20 + 20 = 0
20 + 3x + -1x2 = 0 + x + -1x
20 + 3x + -1x2 = x + -1x

Combine like terms: x + -1x = 0
20 + 3x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-20 + -3x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -3x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + -3x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -3x + x2 = 0 + 20
-3x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
-3x + x2 = 20

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 20 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 20 + 2.25

Combine like terms: 20 + 2.25 = 22.25
2.25 + -3x + x2 = 22.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(x + -1.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

x + -1.5 = 4.716990566 Simplifying x + -1.5 = 4.716990566 Reorder the terms: -1.5 + x = 4.716990566 Solving -1.5 + x = 4.716990566 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 4.716990566 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 4.716990566 + 1.5 x = 4.716990566 + 1.5 Combine like terms: 4.716990566 + 1.5 = 6.216990566 x = 6.216990566 Simplifying x = 6.216990566

Subproblem 2

x + -1.5 = -4.716990566 Simplifying x + -1.5 = -4.716990566 Reorder the terms: -1.5 + x = -4.716990566 Solving -1.5 + x = -4.716990566 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -4.716990566 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -4.716990566 + 1.5 x = -4.716990566 + 1.5 Combine like terms: -4.716990566 + 1.5 = -3.216990566 x = -3.216990566 Simplifying x = -3.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.216990566, -3.216990566}

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