0.1115898-1.3692x+4.2x^2=x^2

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Solution for 0.1115898-1.3692x+4.2x^2=x^2 equation:


Simplifying
0.1115898 + -1.3692x + 4.2x2 = x2

Solving
0.1115898 + -1.3692x + 4.2x2 = x2

Solving for variable 'x'.

Combine like terms: 4.2x2 + -1x2 = 3.2x2
0.1115898 + -1.3692x + 3.2x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
0.1115898 + -1.3692x + 3.2x2 = 0

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

Divide each side by '3.2'.
0.0348718125 + -0.427875x + x2 = 0

Move the constant term to the right:

Add '-0.0348718125' to each side of the equation.
0.0348718125 + -0.427875x + -0.0348718125 + x2 = 0 + -0.0348718125

Reorder the terms:
0.0348718125 + -0.0348718125 + -0.427875x + x2 = 0 + -0.0348718125

Combine like terms: 0.0348718125 + -0.0348718125 = 0.0000000000
0.0000000000 + -0.427875x + x2 = 0 + -0.0348718125
-0.427875x + x2 = 0 + -0.0348718125

Combine like terms: 0 + -0.0348718125 = -0.0348718125
-0.427875x + x2 = -0.0348718125

The x term is -0.427875x.  Take half its coefficient (-0.2139375).
Square it (0.04576925391) and add it to both sides.

Add '0.04576925391' to each side of the equation.
-0.427875x + 0.04576925391 + x2 = -0.0348718125 + 0.04576925391

Reorder the terms:
0.04576925391 + -0.427875x + x2 = -0.0348718125 + 0.04576925391

Combine like terms: -0.0348718125 + 0.04576925391 = 0.01089744141
0.04576925391 + -0.427875x + x2 = 0.01089744141

Factor a perfect square on the left side:
(x + -0.2139375)(x + -0.2139375) = 0.01089744141

Calculate the square root of the right side: 0.104390811

Break this problem into two subproblems by setting 
(x + -0.2139375) equal to 0.104390811 and -0.104390811.

Subproblem 1

x + -0.2139375 = 0.104390811 Simplifying x + -0.2139375 = 0.104390811 Reorder the terms: -0.2139375 + x = 0.104390811 Solving -0.2139375 + x = 0.104390811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2139375' to each side of the equation. -0.2139375 + 0.2139375 + x = 0.104390811 + 0.2139375 Combine like terms: -0.2139375 + 0.2139375 = 0.0000000 0.0000000 + x = 0.104390811 + 0.2139375 x = 0.104390811 + 0.2139375 Combine like terms: 0.104390811 + 0.2139375 = 0.318328311 x = 0.318328311 Simplifying x = 0.318328311

Subproblem 2

x + -0.2139375 = -0.104390811 Simplifying x + -0.2139375 = -0.104390811 Reorder the terms: -0.2139375 + x = -0.104390811 Solving -0.2139375 + x = -0.104390811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2139375' to each side of the equation. -0.2139375 + 0.2139375 + x = -0.104390811 + 0.2139375 Combine like terms: -0.2139375 + 0.2139375 = 0.0000000 0.0000000 + x = -0.104390811 + 0.2139375 x = -0.104390811 + 0.2139375 Combine like terms: -0.104390811 + 0.2139375 = 0.109546689 x = 0.109546689 Simplifying x = 0.109546689

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.318328311, 0.109546689}

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