-x^2+8x=-31

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Solution for -x^2+8x=-31 equation:


Simplifying
-1x2 + 8x = -31

Reorder the terms:
8x + -1x2 = -31

Solving
8x + -1x2 = -31

Solving for variable 'x'.

Reorder the terms:
31 + 8x + -1x2 = -31 + 31

Combine like terms: -31 + 31 = 0
31 + 8x + -1x2 = 0

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

Divide each side by '-1'.
-31 + -8x + x2 = 0

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -8x + 31 + x2 = 0 + 31

Reorder the terms:
-31 + 31 + -8x + x2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -8x + x2 = 0 + 31
-8x + x2 = 0 + 31

Combine like terms: 0 + 31 = 31
-8x + x2 = 31

The x term is -8x.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8x + 16 + x2 = 31 + 16

Reorder the terms:
16 + -8x + x2 = 31 + 16

Combine like terms: 31 + 16 = 47
16 + -8x + x2 = 47

Factor a perfect square on the left side:
(x + -4)(x + -4) = 47

Calculate the square root of the right side: 6.8556546

Break this problem into two subproblems by setting 
(x + -4) equal to 6.8556546 and -6.8556546.

Subproblem 1

x + -4 = 6.8556546 Simplifying x + -4 = 6.8556546 Reorder the terms: -4 + x = 6.8556546 Solving -4 + x = 6.8556546 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = 6.8556546 + 4 Combine like terms: -4 + 4 = 0 0 + x = 6.8556546 + 4 x = 6.8556546 + 4 Combine like terms: 6.8556546 + 4 = 10.8556546 x = 10.8556546 Simplifying x = 10.8556546

Subproblem 2

x + -4 = -6.8556546 Simplifying x + -4 = -6.8556546 Reorder the terms: -4 + x = -6.8556546 Solving -4 + x = -6.8556546 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + x = -6.8556546 + 4 Combine like terms: -4 + 4 = 0 0 + x = -6.8556546 + 4 x = -6.8556546 + 4 Combine like terms: -6.8556546 + 4 = -2.8556546 x = -2.8556546 Simplifying x = -2.8556546

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.8556546, -2.8556546}

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