x^2=7x-1

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Solution for x^2=7x-1 equation:


Simplifying
x2 = 7x + -1

Reorder the terms:
x2 = -1 + 7x

Solving
x2 = -1 + 7x

Solving for variable 'x'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + -7x + x2 = 0 + -1

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

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

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -1 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + -7x + x2 = 11.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1

x + -3.5 = 3.354101966 Simplifying x + -3.5 = 3.354101966 Reorder the terms: -3.5 + x = 3.354101966 Solving -3.5 + x = 3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 3.354101966 + 3.5 x = 3.354101966 + 3.5 Combine like terms: 3.354101966 + 3.5 = 6.854101966 x = 6.854101966 Simplifying x = 6.854101966

Subproblem 2

x + -3.5 = -3.354101966 Simplifying x + -3.5 = -3.354101966 Reorder the terms: -3.5 + x = -3.354101966 Solving -3.5 + x = -3.354101966 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -3.354101966 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -3.354101966 + 3.5 x = -3.354101966 + 3.5 Combine like terms: -3.354101966 + 3.5 = 0.145898034 x = 0.145898034 Simplifying x = 0.145898034

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.854101966, 0.145898034}

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