x^2=7x-7

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


Simplifying
x2 = 7x + -7

Reorder the terms:
x2 = -7 + 7x

Solving
x2 = -7 + 7x

Solving for variable 'x'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = -7 + 12.25

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

Combine like terms: -7 + 12.25 = 5.25
12.25 + -7x + x2 = 5.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -3.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

x + -3.5 = 2.291287847 Simplifying x + -3.5 = 2.291287847 Reorder the terms: -3.5 + x = 2.291287847 Solving -3.5 + x = 2.291287847 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 = 2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 2.291287847 + 3.5 x = 2.291287847 + 3.5 Combine like terms: 2.291287847 + 3.5 = 5.791287847 x = 5.791287847 Simplifying x = 5.791287847

Subproblem 2

x + -3.5 = -2.291287847 Simplifying x + -3.5 = -2.291287847 Reorder the terms: -3.5 + x = -2.291287847 Solving -3.5 + x = -2.291287847 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 = -2.291287847 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -2.291287847 + 3.5 x = -2.291287847 + 3.5 Combine like terms: -2.291287847 + 3.5 = 1.208712153 x = 1.208712153 Simplifying x = 1.208712153

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.791287847, 1.208712153}

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