3x^2=10x+3

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


Simplifying
3x2 = 10x + 3

Reorder the terms:
3x2 = 3 + 10x

Solving
3x2 = 3 + 10x

Solving for variable 'x'.

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

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

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

Combine like terms: 10x + -10x = 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 = 3.610317299 x = 3.610317299 Simplifying x = 3.610317299

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 = -0.276983965 x = -0.276983965 Simplifying x = -0.276983965

Solution

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

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