0=x^2-14x-84

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Solution for 0=x^2-14x-84 equation:


Simplifying
0 = x2 + -14x + -84

Reorder the terms:
0 = -84 + -14x + x2

Solving
0 = -84 + -14x + x2

Solving for variable 'x'.

Combine like terms: 0 + 84 = 84
84 + 14x + -1x2 = -84 + -14x + x2 + 84 + 14x + -1x2

Reorder the terms:
84 + 14x + -1x2 = -84 + 84 + -14x + 14x + x2 + -1x2

Combine like terms: -84 + 84 = 0
84 + 14x + -1x2 = 0 + -14x + 14x + x2 + -1x2
84 + 14x + -1x2 = -14x + 14x + x2 + -1x2

Combine like terms: -14x + 14x = 0
84 + 14x + -1x2 = 0 + x2 + -1x2
84 + 14x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
84 + 14x + -1x2 = 0

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

Divide each side by '-1'.
-84 + -14x + x2 = 0

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + -14x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + -14x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + -14x + x2 = 0 + 84
-14x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
-14x + x2 = 84

The x term is -14x.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14x + 49 + x2 = 84 + 49

Reorder the terms:
49 + -14x + x2 = 84 + 49

Combine like terms: 84 + 49 = 133
49 + -14x + x2 = 133

Factor a perfect square on the left side:
(x + -7)(x + -7) = 133

Calculate the square root of the right side: 11.532562595

Break this problem into two subproblems by setting 
(x + -7) equal to 11.532562595 and -11.532562595.

Subproblem 1

x + -7 = 11.532562595 Simplifying x + -7 = 11.532562595 Reorder the terms: -7 + x = 11.532562595 Solving -7 + x = 11.532562595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + x = 11.532562595 + 7 Combine like terms: -7 + 7 = 0 0 + x = 11.532562595 + 7 x = 11.532562595 + 7 Combine like terms: 11.532562595 + 7 = 18.532562595 x = 18.532562595 Simplifying x = 18.532562595

Subproblem 2

x + -7 = -11.532562595 Simplifying x + -7 = -11.532562595 Reorder the terms: -7 + x = -11.532562595 Solving -7 + x = -11.532562595 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + x = -11.532562595 + 7 Combine like terms: -7 + 7 = 0 0 + x = -11.532562595 + 7 x = -11.532562595 + 7 Combine like terms: -11.532562595 + 7 = -4.532562595 x = -4.532562595 Simplifying x = -4.532562595

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.532562595, -4.532562595}

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