-x^2=13x-42

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Solution for -x^2=13x-42 equation:


Simplifying
-1x2 = 13x + -42

Reorder the terms:
-1x2 = -42 + 13x

Solving
-1x2 = -42 + 13x

Solving for variable 'x'.

Reorder the terms:
42 + -13x + -1x2 = -42 + 13x + 42 + -13x

Reorder the terms:
42 + -13x + -1x2 = -42 + 42 + 13x + -13x

Combine like terms: -42 + 42 = 0
42 + -13x + -1x2 = 0 + 13x + -13x
42 + -13x + -1x2 = 13x + -13x

Combine like terms: 13x + -13x = 0
42 + -13x + -1x2 = 0

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

Divide each side by '-1'.
-42 + 13x + x2 = 0

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + 13x + 42 + x2 = 0 + 42

Reorder the terms:
-42 + 42 + 13x + x2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + 13x + x2 = 0 + 42
13x + x2 = 0 + 42

Combine like terms: 0 + 42 = 42
13x + x2 = 42

The x term is 13x.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13x + 42.25 + x2 = 42 + 42.25

Reorder the terms:
42.25 + 13x + x2 = 42 + 42.25

Combine like terms: 42 + 42.25 = 84.25
42.25 + 13x + x2 = 84.25

Factor a perfect square on the left side:
(x + 6.5)(x + 6.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(x + 6.5) equal to 9.178779875 and -9.178779875.

Subproblem 1

x + 6.5 = 9.178779875 Simplifying x + 6.5 = 9.178779875 Reorder the terms: 6.5 + x = 9.178779875 Solving 6.5 + x = 9.178779875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = 9.178779875 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 9.178779875 + -6.5 x = 9.178779875 + -6.5 Combine like terms: 9.178779875 + -6.5 = 2.678779875 x = 2.678779875 Simplifying x = 2.678779875

Subproblem 2

x + 6.5 = -9.178779875 Simplifying x + 6.5 = -9.178779875 Reorder the terms: 6.5 + x = -9.178779875 Solving 6.5 + x = -9.178779875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = -9.178779875 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -9.178779875 + -6.5 x = -9.178779875 + -6.5 Combine like terms: -9.178779875 + -6.5 = -15.678779875 x = -15.678779875 Simplifying x = -15.678779875

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.678779875, -15.678779875}

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