7x-5=x^2

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Solution for 7x-5=x^2 equation:


Simplifying
7x + -5 = x2

Reorder the terms:
-5 + 7x = x2

Solving
-5 + 7x = x2

Solving for variable 'x'.

Combine like terms: x2 + -1x2 = 0
-5 + 7x + -1x2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
5 + -7x + x2 = 0

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + -7x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + -7x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -7x + x2 = 0 + -5
-7x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
-7x + x2 = -5

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 = -5 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -5 + 12.25

Combine like terms: -5 + 12.25 = 7.25
12.25 + -7x + x2 = 7.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(x + -3.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

x + -3.5 = 2.692582404 Simplifying x + -3.5 = 2.692582404 Reorder the terms: -3.5 + x = 2.692582404 Solving -3.5 + x = 2.692582404 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 = 2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 2.692582404 + 3.5 x = 2.692582404 + 3.5 Combine like terms: 2.692582404 + 3.5 = 6.192582404 x = 6.192582404 Simplifying x = 6.192582404

Subproblem 2

x + -3.5 = -2.692582404 Simplifying x + -3.5 = -2.692582404 Reorder the terms: -3.5 + x = -2.692582404 Solving -3.5 + x = -2.692582404 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 = -2.692582404 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -2.692582404 + 3.5 x = -2.692582404 + 3.5 Combine like terms: -2.692582404 + 3.5 = 0.807417596 x = 0.807417596 Simplifying x = 0.807417596

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.192582404, 0.807417596}

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