.5(x-6)+1=2x(x-10)-3

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Solution for .5(x-6)+1=2x(x-10)-3 equation:


Simplifying
0.5(x + -6) + 1 = 2x(x + -10) + -3

Reorder the terms:
0.5(-6 + x) + 1 = 2x(x + -10) + -3
(-6 * 0.5 + x * 0.5) + 1 = 2x(x + -10) + -3
(-3 + 0.5x) + 1 = 2x(x + -10) + -3

Reorder the terms:
-3 + 1 + 0.5x = 2x(x + -10) + -3

Combine like terms: -3 + 1 = -2
-2 + 0.5x = 2x(x + -10) + -3

Reorder the terms:
-2 + 0.5x = 2x(-10 + x) + -3
-2 + 0.5x = (-10 * 2x + x * 2x) + -3
-2 + 0.5x = (-20x + 2x2) + -3

Reorder the terms:
-2 + 0.5x = -3 + -20x + 2x2

Solving
-2 + 0.5x = -3 + -20x + 2x2

Solving for variable 'x'.

Reorder the terms:
-2 + 3 + 0.5x + 20x + -2x2 = -3 + -20x + 2x2 + 3 + 20x + -2x2

Combine like terms: -2 + 3 = 1
1 + 0.5x + 20x + -2x2 = -3 + -20x + 2x2 + 3 + 20x + -2x2

Combine like terms: 0.5x + 20x = 20.5x
1 + 20.5x + -2x2 = -3 + -20x + 2x2 + 3 + 20x + -2x2

Reorder the terms:
1 + 20.5x + -2x2 = -3 + 3 + -20x + 20x + 2x2 + -2x2

Combine like terms: -3 + 3 = 0
1 + 20.5x + -2x2 = 0 + -20x + 20x + 2x2 + -2x2
1 + 20.5x + -2x2 = -20x + 20x + 2x2 + -2x2

Combine like terms: -20x + 20x = 0
1 + 20.5x + -2x2 = 0 + 2x2 + -2x2
1 + 20.5x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
1 + 20.5x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-0.5 + -10.25x + x2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -10.25x + 0.5 + x2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -10.25x + x2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -10.25x + x2 = 0 + 0.5
-10.25x + x2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-10.25x + x2 = 0.5

The x term is -10.25x.  Take half its coefficient (-5.125).
Square it (26.265625) and add it to both sides.

Add '26.265625' to each side of the equation.
-10.25x + 26.265625 + x2 = 0.5 + 26.265625

Reorder the terms:
26.265625 + -10.25x + x2 = 0.5 + 26.265625

Combine like terms: 0.5 + 26.265625 = 26.765625
26.265625 + -10.25x + x2 = 26.765625

Factor a perfect square on the left side:
(x + -5.125)(x + -5.125) = 26.765625

Calculate the square root of the right side: 5.173550522

Break this problem into two subproblems by setting 
(x + -5.125) equal to 5.173550522 and -5.173550522.

Subproblem 1

x + -5.125 = 5.173550522 Simplifying x + -5.125 = 5.173550522 Reorder the terms: -5.125 + x = 5.173550522 Solving -5.125 + x = 5.173550522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + x = 5.173550522 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + x = 5.173550522 + 5.125 x = 5.173550522 + 5.125 Combine like terms: 5.173550522 + 5.125 = 10.298550522 x = 10.298550522 Simplifying x = 10.298550522

Subproblem 2

x + -5.125 = -5.173550522 Simplifying x + -5.125 = -5.173550522 Reorder the terms: -5.125 + x = -5.173550522 Solving -5.125 + x = -5.173550522 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.125' to each side of the equation. -5.125 + 5.125 + x = -5.173550522 + 5.125 Combine like terms: -5.125 + 5.125 = 0.000 0.000 + x = -5.173550522 + 5.125 x = -5.173550522 + 5.125 Combine like terms: -5.173550522 + 5.125 = -0.048550522 x = -0.048550522 Simplifying x = -0.048550522

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.298550522, -0.048550522}

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