x=x^2-2x-3

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Solution for x=x^2-2x-3 equation:


Simplifying
x = x2 + -2x + -3

Reorder the terms:
x = -3 + -2x + x2

Solving
x = -3 + -2x + x2

Solving for variable 'x'.

Reorder the terms:
3 + x + 2x + -1x2 = -3 + -2x + x2 + 3 + 2x + -1x2

Combine like terms: x + 2x = 3x
3 + 3x + -1x2 = -3 + -2x + x2 + 3 + 2x + -1x2

Reorder the terms:
3 + 3x + -1x2 = -3 + 3 + -2x + 2x + x2 + -1x2

Combine like terms: -3 + 3 = 0
3 + 3x + -1x2 = 0 + -2x + 2x + x2 + -1x2
3 + 3x + -1x2 = -2x + 2x + x2 + -1x2

Combine like terms: -2x + 2x = 0
3 + 3x + -1x2 = 0 + x2 + -1x2
3 + 3x + -1x2 = x2 + -1x2

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

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

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

Move the constant term to the right:

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

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

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

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

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 = 3 + 2.25

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

Combine like terms: 3 + 2.25 = 5.25
2.25 + -3x + x2 = 5.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

x + -1.5 = 2.291287847 Simplifying x + -1.5 = 2.291287847 Reorder the terms: -1.5 + x = 2.291287847 Solving -1.5 + x = 2.291287847 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 = 2.291287847 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 2.291287847 + 1.5 x = 2.291287847 + 1.5 Combine like terms: 2.291287847 + 1.5 = 3.791287847 x = 3.791287847 Simplifying x = 3.791287847

Subproblem 2

x + -1.5 = -2.291287847 Simplifying x + -1.5 = -2.291287847 Reorder the terms: -1.5 + x = -2.291287847 Solving -1.5 + x = -2.291287847 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 = -2.291287847 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -2.291287847 + 1.5 x = -2.291287847 + 1.5 Combine like terms: -2.291287847 + 1.5 = -0.791287847 x = -0.791287847 Simplifying x = -0.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.791287847, -0.791287847}

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