6x=7x^2+1

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Solution for 6x=7x^2+1 equation:


Simplifying
6x = 7x2 + 1

Reorder the terms:
6x = 1 + 7x2

Solving
6x = 1 + 7x2

Solving for variable 'x'.

Reorder the terms:
-1 + 6x + -7x2 = 1 + 7x2 + -1 + -7x2

Reorder the terms:
-1 + 6x + -7x2 = 1 + -1 + 7x2 + -7x2

Combine like terms: 1 + -1 = 0
-1 + 6x + -7x2 = 0 + 7x2 + -7x2
-1 + 6x + -7x2 = 7x2 + -7x2

Combine like terms: 7x2 + -7x2 = 0
-1 + 6x + -7x2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
0.1428571429 + -0.8571428571x + x2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + -0.8571428571x + -0.1428571429 + x2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + -0.8571428571x + x2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + -0.8571428571x + x2 = 0 + -0.1428571429
-0.8571428571x + x2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
-0.8571428571x + x2 = -0.1428571429

The x term is -0.8571428571x.  Take half its coefficient (-0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
-0.8571428571x + 0.1836734694 + x2 = -0.1428571429 + 0.1836734694

Reorder the terms:
0.1836734694 + -0.8571428571x + x2 = -0.1428571429 + 0.1836734694

Combine like terms: -0.1428571429 + 0.1836734694 = 0.0408163265
0.1836734694 + -0.8571428571x + x2 = 0.0408163265

Factor a perfect square on the left side:
(x + -0.4285714286)(x + -0.4285714286) = 0.0408163265

Calculate the square root of the right side: 0.202030509

Break this problem into two subproblems by setting 
(x + -0.4285714286) equal to 0.202030509 and -0.202030509.

Subproblem 1

x + -0.4285714286 = 0.202030509 Simplifying x + -0.4285714286 = 0.202030509 Reorder the terms: -0.4285714286 + x = 0.202030509 Solving -0.4285714286 + x = 0.202030509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = 0.202030509 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = 0.202030509 + 0.4285714286 x = 0.202030509 + 0.4285714286 Combine like terms: 0.202030509 + 0.4285714286 = 0.6306019376 x = 0.6306019376 Simplifying x = 0.6306019376

Subproblem 2

x + -0.4285714286 = -0.202030509 Simplifying x + -0.4285714286 = -0.202030509 Reorder the terms: -0.4285714286 + x = -0.202030509 Solving -0.4285714286 + x = -0.202030509 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = -0.202030509 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = -0.202030509 + 0.4285714286 x = -0.202030509 + 0.4285714286 Combine like terms: -0.202030509 + 0.4285714286 = 0.2265409196 x = 0.2265409196 Simplifying x = 0.2265409196

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6306019376, 0.2265409196}

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