5x+2=6x^2+7x-3

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Solution for 5x+2=6x^2+7x-3 equation:


Simplifying
5x + 2 = 6x2 + 7x + -3

Reorder the terms:
2 + 5x = 6x2 + 7x + -3

Reorder the terms:
2 + 5x = -3 + 7x + 6x2

Solving
2 + 5x = -3 + 7x + 6x2

Solving for variable 'x'.

Reorder the terms:
2 + 3 + 5x + -7x + -6x2 = -3 + 7x + 6x2 + 3 + -7x + -6x2

Combine like terms: 2 + 3 = 5
5 + 5x + -7x + -6x2 = -3 + 7x + 6x2 + 3 + -7x + -6x2

Combine like terms: 5x + -7x = -2x
5 + -2x + -6x2 = -3 + 7x + 6x2 + 3 + -7x + -6x2

Reorder the terms:
5 + -2x + -6x2 = -3 + 3 + 7x + -7x + 6x2 + -6x2

Combine like terms: -3 + 3 = 0
5 + -2x + -6x2 = 0 + 7x + -7x + 6x2 + -6x2
5 + -2x + -6x2 = 7x + -7x + 6x2 + -6x2

Combine like terms: 7x + -7x = 0
5 + -2x + -6x2 = 0 + 6x2 + -6x2
5 + -2x + -6x2 = 6x2 + -6x2

Combine like terms: 6x2 + -6x2 = 0
5 + -2x + -6x2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-0.8333333333 + 0.3333333333x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 0.3333333333x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 0.3333333333x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 0.3333333333x + x2 = 0 + 0.8333333333
0.3333333333x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
0.3333333333x + x2 = 0.8333333333

The x term is 0.3333333333x.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333x + 0.02777777779 + x2 = 0.8333333333 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333x + x2 = 0.8333333333 + 0.02777777779

Combine like terms: 0.8333333333 + 0.02777777779 = 0.86111111109
0.02777777779 + 0.3333333333x + x2 = 0.86111111109

Factor a perfect square on the left side:
(x + 0.1666666667)(x + 0.1666666667) = 0.86111111109

Calculate the square root of the right side: 0.927960727

Break this problem into two subproblems by setting 
(x + 0.1666666667) equal to 0.927960727 and -0.927960727.

Subproblem 1

x + 0.1666666667 = 0.927960727 Simplifying x + 0.1666666667 = 0.927960727 Reorder the terms: 0.1666666667 + x = 0.927960727 Solving 0.1666666667 + x = 0.927960727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.927960727 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.927960727 + -0.1666666667 x = 0.927960727 + -0.1666666667 Combine like terms: 0.927960727 + -0.1666666667 = 0.7612940603 x = 0.7612940603 Simplifying x = 0.7612940603

Subproblem 2

x + 0.1666666667 = -0.927960727 Simplifying x + 0.1666666667 = -0.927960727 Reorder the terms: 0.1666666667 + x = -0.927960727 Solving 0.1666666667 + x = -0.927960727 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.927960727 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.927960727 + -0.1666666667 x = -0.927960727 + -0.1666666667 Combine like terms: -0.927960727 + -0.1666666667 = -1.0946273937 x = -1.0946273937 Simplifying x = -1.0946273937

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.7612940603, -1.0946273937}

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