0.5x^2-0.5x-78=0

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Solution for 0.5x^2-0.5x-78=0 equation:


Simplifying
0.5x2 + -0.5x + -78 = 0

Reorder the terms:
-78 + -0.5x + 0.5x2 = 0

Solving
-78 + -0.5x + 0.5x2 = 0

Solving for variable 'x'.

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

Divide each side by '0.5'.
-156 + -1x + x2 = 0

Move the constant term to the right:

Add '156' to each side of the equation.
-156 + -1x + 156 + x2 = 0 + 156

Reorder the terms:
-156 + 156 + -1x + x2 = 0 + 156

Combine like terms: -156 + 156 = 0
0 + -1x + x2 = 0 + 156
-1x + x2 = 0 + 156

Combine like terms: 0 + 156 = 156
-1x + x2 = 156

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 156 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 156 + 0.25

Combine like terms: 156 + 0.25 = 156.25
0.25 + -1x + x2 = 156.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 156.25

Calculate the square root of the right side: 12.5

Break this problem into two subproblems by setting 
(x + -0.5) equal to 12.5 and -12.5.

Subproblem 1

x + -0.5 = 12.5 Simplifying x + -0.5 = 12.5 Reorder the terms: -0.5 + x = 12.5 Solving -0.5 + x = 12.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 12.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 12.5 + 0.5 x = 12.5 + 0.5 Combine like terms: 12.5 + 0.5 = 13 x = 13 Simplifying x = 13

Subproblem 2

x + -0.5 = -12.5 Simplifying x + -0.5 = -12.5 Reorder the terms: -0.5 + x = -12.5 Solving -0.5 + x = -12.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -12.5 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -12.5 + 0.5 x = -12.5 + 0.5 Combine like terms: -12.5 + 0.5 = -12 x = -12 Simplifying x = -12

Solution

The solution to the problem is based on the solutions from the subproblems. x = {13, -12}

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