x^2+3=10x+4x^2

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Solution for x^2+3=10x+4x^2 equation:


Simplifying
x2 + 3 = 10x + 4x2

Reorder the terms:
3 + x2 = 10x + 4x2

Solving
3 + x2 = 10x + 4x2

Solving for variable 'x'.

Reorder the terms:
3 + -10x + x2 + -4x2 = 10x + 4x2 + -10x + -4x2

Combine like terms: x2 + -4x2 = -3x2
3 + -10x + -3x2 = 10x + 4x2 + -10x + -4x2

Reorder the terms:
3 + -10x + -3x2 = 10x + -10x + 4x2 + -4x2

Combine like terms: 10x + -10x = 0
3 + -10x + -3x2 = 0 + 4x2 + -4x2
3 + -10x + -3x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
3 + -10x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-1 + 3.333333333x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 3.333333333x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 3.333333333x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 3.333333333x + x2 = 0 + 1
3.333333333x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
3.333333333x + x2 = 1

The x term is 3.333333333x.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333x + 2.777777779 + x2 = 1 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333x + x2 = 1 + 2.777777779

Combine like terms: 1 + 2.777777779 = 3.777777779
2.777777779 + 3.333333333x + x2 = 3.777777779

Factor a perfect square on the left side:
(x + 1.666666667)(x + 1.666666667) = 3.777777779

Calculate the square root of the right side: 1.943650632

Break this problem into two subproblems by setting 
(x + 1.666666667) equal to 1.943650632 and -1.943650632.

Subproblem 1

x + 1.666666667 = 1.943650632 Simplifying x + 1.666666667 = 1.943650632 Reorder the terms: 1.666666667 + x = 1.943650632 Solving 1.666666667 + x = 1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 1.943650632 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 1.943650632 + -1.666666667 x = 1.943650632 + -1.666666667 Combine like terms: 1.943650632 + -1.666666667 = 0.276983965 x = 0.276983965 Simplifying x = 0.276983965

Subproblem 2

x + 1.666666667 = -1.943650632 Simplifying x + 1.666666667 = -1.943650632 Reorder the terms: 1.666666667 + x = -1.943650632 Solving 1.666666667 + x = -1.943650632 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -1.943650632 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -1.943650632 + -1.666666667 x = -1.943650632 + -1.666666667 Combine like terms: -1.943650632 + -1.666666667 = -3.610317299 x = -3.610317299 Simplifying x = -3.610317299

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.276983965, -3.610317299}

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