0=x^2-7x-312

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Solution for 0=x^2-7x-312 equation:


Simplifying
0 = x2 + -7x + -312

Reorder the terms:
0 = -312 + -7x + x2

Solving
0 = -312 + -7x + x2

Solving for variable 'x'.

Combine like terms: 0 + 312 = 312
312 + 7x + -1x2 = -312 + -7x + x2 + 312 + 7x + -1x2

Reorder the terms:
312 + 7x + -1x2 = -312 + 312 + -7x + 7x + x2 + -1x2

Combine like terms: -312 + 312 = 0
312 + 7x + -1x2 = 0 + -7x + 7x + x2 + -1x2
312 + 7x + -1x2 = -7x + 7x + x2 + -1x2

Combine like terms: -7x + 7x = 0
312 + 7x + -1x2 = 0 + x2 + -1x2
312 + 7x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
312 + 7x + -1x2 = 0

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

Divide each side by '-1'.
-312 + -7x + x2 = 0

Move the constant term to the right:

Add '312' to each side of the equation.
-312 + -7x + 312 + x2 = 0 + 312

Reorder the terms:
-312 + 312 + -7x + x2 = 0 + 312

Combine like terms: -312 + 312 = 0
0 + -7x + x2 = 0 + 312
-7x + x2 = 0 + 312

Combine like terms: 0 + 312 = 312
-7x + x2 = 312

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 = 312 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 312 + 12.25

Combine like terms: 312 + 12.25 = 324.25
12.25 + -7x + x2 = 324.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 324.25

Calculate the square root of the right side: 18.006943105

Break this problem into two subproblems by setting 
(x + -3.5) equal to 18.006943105 and -18.006943105.

Subproblem 1

x + -3.5 = 18.006943105 Simplifying x + -3.5 = 18.006943105 Reorder the terms: -3.5 + x = 18.006943105 Solving -3.5 + x = 18.006943105 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 = 18.006943105 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 18.006943105 + 3.5 x = 18.006943105 + 3.5 Combine like terms: 18.006943105 + 3.5 = 21.506943105 x = 21.506943105 Simplifying x = 21.506943105

Subproblem 2

x + -3.5 = -18.006943105 Simplifying x + -3.5 = -18.006943105 Reorder the terms: -3.5 + x = -18.006943105 Solving -3.5 + x = -18.006943105 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 = -18.006943105 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -18.006943105 + 3.5 x = -18.006943105 + 3.5 Combine like terms: -18.006943105 + 3.5 = -14.506943105 x = -14.506943105 Simplifying x = -14.506943105

Solution

The solution to the problem is based on the solutions from the subproblems. x = {21.506943105, -14.506943105}

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