7x^2-x=7

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Solution for 7x^2-x=7 equation:


Simplifying
7x2 + -1x = 7

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

Solving
-1x + 7x2 = 7

Solving for variable 'x'.

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

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

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

Divide each side by '7'.
-1 + -0.1428571429x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.1428571429x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.1428571429x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.1428571429x + x2 = 0 + 1
-0.1428571429x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.1428571429x + x2 = 1

The x term is -0.1428571429x.  Take half its coefficient (-0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
-0.1428571429x + 0.005102040819 + x2 = 1 + 0.005102040819

Reorder the terms:
0.005102040819 + -0.1428571429x + x2 = 1 + 0.005102040819

Combine like terms: 1 + 0.005102040819 = 1.005102040819
0.005102040819 + -0.1428571429x + x2 = 1.005102040819

Factor a perfect square on the left side:
(x + -0.07142857145)(x + -0.07142857145) = 1.005102040819

Calculate the square root of the right side: 1.002547775

Break this problem into two subproblems by setting 
(x + -0.07142857145) equal to 1.002547775 and -1.002547775.

Subproblem 1

x + -0.07142857145 = 1.002547775 Simplifying x + -0.07142857145 = 1.002547775 Reorder the terms: -0.07142857145 + x = 1.002547775 Solving -0.07142857145 + x = 1.002547775 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + x = 1.002547775 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + x = 1.002547775 + 0.07142857145 x = 1.002547775 + 0.07142857145 Combine like terms: 1.002547775 + 0.07142857145 = 1.07397634645 x = 1.07397634645 Simplifying x = 1.07397634645

Subproblem 2

x + -0.07142857145 = -1.002547775 Simplifying x + -0.07142857145 = -1.002547775 Reorder the terms: -0.07142857145 + x = -1.002547775 Solving -0.07142857145 + x = -1.002547775 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.07142857145' to each side of the equation. -0.07142857145 + 0.07142857145 + x = -1.002547775 + 0.07142857145 Combine like terms: -0.07142857145 + 0.07142857145 = 0.00000000000 0.00000000000 + x = -1.002547775 + 0.07142857145 x = -1.002547775 + 0.07142857145 Combine like terms: -1.002547775 + 0.07142857145 = -0.93111920355 x = -0.93111920355 Simplifying x = -0.93111920355

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.07397634645, -0.93111920355}

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