6x^2=7x+12

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Solution for 6x^2=7x+12 equation:


Simplifying
6x2 = 7x + 12

Reorder the terms:
6x2 = 12 + 7x

Solving
6x2 = 12 + 7x

Solving for variable 'x'.

Reorder the terms:
-12 + -7x + 6x2 = 12 + 7x + -12 + -7x

Reorder the terms:
-12 + -7x + 6x2 = 12 + -12 + 7x + -7x

Combine like terms: 12 + -12 = 0
-12 + -7x + 6x2 = 0 + 7x + -7x
-12 + -7x + 6x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-12 + -7x + 6x2 = 0

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

Divide each side by '6'.
-2 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -1.166666667x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -1.166666667x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -1.166666667x + x2 = 0 + 2
-1.166666667x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-1.166666667x + x2 = 2

The x term is -1.166666667x.  Take half its coefficient (-0.5833333335).
Square it (0.3402777780) and add it to both sides.

Add '0.3402777780' to each side of the equation.
-1.166666667x + 0.3402777780 + x2 = 2 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 2 + 0.3402777780

Combine like terms: 2 + 0.3402777780 = 2.340277778
0.3402777780 + -1.166666667x + x2 = 2.340277778

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 2.340277778

Calculate the square root of the right side: 1.529796646

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 1.529796646 and -1.529796646.

Subproblem 1

x + -0.5833333335 = 1.529796646 Simplifying x + -0.5833333335 = 1.529796646 Reorder the terms: -0.5833333335 + x = 1.529796646 Solving -0.5833333335 + x = 1.529796646 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = 1.529796646 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.529796646 + 0.5833333335 x = 1.529796646 + 0.5833333335 Combine like terms: 1.529796646 + 0.5833333335 = 2.1131299795 x = 2.1131299795 Simplifying x = 2.1131299795

Subproblem 2

x + -0.5833333335 = -1.529796646 Simplifying x + -0.5833333335 = -1.529796646 Reorder the terms: -0.5833333335 + x = -1.529796646 Solving -0.5833333335 + x = -1.529796646 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5833333335' to each side of the equation. -0.5833333335 + 0.5833333335 + x = -1.529796646 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.529796646 + 0.5833333335 x = -1.529796646 + 0.5833333335 Combine like terms: -1.529796646 + 0.5833333335 = -0.9464633125 x = -0.9464633125 Simplifying x = -0.9464633125

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.1131299795, -0.9464633125}

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