x(x-3)-(x-2)2=3

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Solution for x(x-3)-(x-2)2=3 equation:


Simplifying
x(x + -3) + -1(x + -2) * 2 = 3

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

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

Reorder the terms for easier multiplication:
-3x + x2 + -1 * 2(-2 + x) = 3

Multiply -1 * 2
-3x + x2 + -2(-2 + x) = 3
-3x + x2 + (-2 * -2 + x * -2) = 3
-3x + x2 + (4 + -2x) = 3

Reorder the terms:
4 + -3x + -2x + x2 = 3

Combine like terms: -3x + -2x = -5x
4 + -5x + x2 = 3

Solving
4 + -5x + x2 = 3

Solving for variable 'x'.

Reorder the terms:
4 + -3 + -5x + x2 = 3 + -3

Combine like terms: 4 + -3 = 1
1 + -5x + x2 = 3 + -3

Combine like terms: 3 + -3 = 0
1 + -5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -5x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -5x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -5x + x2 = 0 + -1
-5x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
-5x + x2 = -1

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -1 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + -5x + x2 = 5.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

x + -2.5 = 2.291287847 Simplifying x + -2.5 = 2.291287847 Reorder the terms: -2.5 + x = 2.291287847 Solving -2.5 + x = 2.291287847 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.291287847 + 2.5 x = 2.291287847 + 2.5 Combine like terms: 2.291287847 + 2.5 = 4.791287847 x = 4.791287847 Simplifying x = 4.791287847

Subproblem 2

x + -2.5 = -2.291287847 Simplifying x + -2.5 = -2.291287847 Reorder the terms: -2.5 + x = -2.291287847 Solving -2.5 + x = -2.291287847 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.291287847 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.291287847 + 2.5 x = -2.291287847 + 2.5 Combine like terms: -2.291287847 + 2.5 = 0.208712153 x = 0.208712153 Simplifying x = 0.208712153

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.791287847, 0.208712153}

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