6x^2=7x+8

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Solution for 6x^2=7x+8 equation:


Simplifying
6x2 = 7x + 8

Reorder the terms:
6x2 = 8 + 7x

Solving
6x2 = 8 + 7x

Solving for variable 'x'.

Reorder the terms:
-8 + -7x + 6x2 = 8 + 7x + -8 + -7x

Reorder the terms:
-8 + -7x + 6x2 = 8 + -8 + 7x + -7x

Combine like terms: 8 + -8 = 0
-8 + -7x + 6x2 = 0 + 7x + -7x
-8 + -7x + 6x2 = 7x + -7x

Combine like terms: 7x + -7x = 0
-8 + -7x + 6x2 = 0

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

Divide each side by '6'.
-1.333333333 + -1.166666667x + x2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + -1.166666667x + 1.333333333 + x2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + -1.166666667x + x2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + -1.166666667x + x2 = 0 + 1.333333333
-1.166666667x + x2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
-1.166666667x + x2 = 1.333333333

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 = 1.333333333 + 0.3402777780

Reorder the terms:
0.3402777780 + -1.166666667x + x2 = 1.333333333 + 0.3402777780

Combine like terms: 1.333333333 + 0.3402777780 = 1.673611111
0.3402777780 + -1.166666667x + x2 = 1.673611111

Factor a perfect square on the left side:
(x + -0.5833333335)(x + -0.5833333335) = 1.673611111

Calculate the square root of the right side: 1.293681225

Break this problem into two subproblems by setting 
(x + -0.5833333335) equal to 1.293681225 and -1.293681225.

Subproblem 1

x + -0.5833333335 = 1.293681225 Simplifying x + -0.5833333335 = 1.293681225 Reorder the terms: -0.5833333335 + x = 1.293681225 Solving -0.5833333335 + x = 1.293681225 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 = 1.293681225 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = 1.293681225 + 0.5833333335 x = 1.293681225 + 0.5833333335 Combine like terms: 1.293681225 + 0.5833333335 = 1.8770145585 x = 1.8770145585 Simplifying x = 1.8770145585

Subproblem 2

x + -0.5833333335 = -1.293681225 Simplifying x + -0.5833333335 = -1.293681225 Reorder the terms: -0.5833333335 + x = -1.293681225 Solving -0.5833333335 + x = -1.293681225 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 = -1.293681225 + 0.5833333335 Combine like terms: -0.5833333335 + 0.5833333335 = 0.0000000000 0.0000000000 + x = -1.293681225 + 0.5833333335 x = -1.293681225 + 0.5833333335 Combine like terms: -1.293681225 + 0.5833333335 = -0.7103478915 x = -0.7103478915 Simplifying x = -0.7103478915

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.8770145585, -0.7103478915}

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