8x^2=-16x+31

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Solution for 8x^2=-16x+31 equation:


Simplifying
8x2 = -16x + 31

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

Solving
8x2 = 31 + -16x

Solving for variable 'x'.

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

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

Combine like terms: 31 + -31 = 0
-31 + 16x + 8x2 = 0 + -16x + 16x
-31 + 16x + 8x2 = -16x + 16x

Combine like terms: -16x + 16x = 0
-31 + 16x + 8x2 = 0

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

Divide each side by '8'.
-3.875 + 2x + x2 = 0

Move the constant term to the right:

Add '3.875' to each side of the equation.
-3.875 + 2x + 3.875 + x2 = 0 + 3.875

Reorder the terms:
-3.875 + 3.875 + 2x + x2 = 0 + 3.875

Combine like terms: -3.875 + 3.875 = 0.000
0.000 + 2x + x2 = 0 + 3.875
2x + x2 = 0 + 3.875

Combine like terms: 0 + 3.875 = 3.875
2x + x2 = 3.875

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 3.875 + 1

Reorder the terms:
1 + 2x + x2 = 3.875 + 1

Combine like terms: 3.875 + 1 = 4.875
1 + 2x + x2 = 4.875

Factor a perfect square on the left side:
(x + 1)(x + 1) = 4.875

Calculate the square root of the right side: 2.207940217

Break this problem into two subproblems by setting 
(x + 1) equal to 2.207940217 and -2.207940217.

Subproblem 1

x + 1 = 2.207940217 Simplifying x + 1 = 2.207940217 Reorder the terms: 1 + x = 2.207940217 Solving 1 + x = 2.207940217 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.207940217 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.207940217 + -1 x = 2.207940217 + -1 Combine like terms: 2.207940217 + -1 = 1.207940217 x = 1.207940217 Simplifying x = 1.207940217

Subproblem 2

x + 1 = -2.207940217 Simplifying x + 1 = -2.207940217 Reorder the terms: 1 + x = -2.207940217 Solving 1 + x = -2.207940217 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.207940217 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.207940217 + -1 x = -2.207940217 + -1 Combine like terms: -2.207940217 + -1 = -3.207940217 x = -3.207940217 Simplifying x = -3.207940217

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.207940217, -3.207940217}

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