0.5(x(x+20))=78

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Solution for 0.5(x(x+20))=78 equation:


Simplifying
0.5(x(x + 20)) = 78

Reorder the terms:
0.5(x(20 + x)) = 78
0.5((20 * x + x * x)) = 78
0.5((20x + x2)) = 78
(20x * 0.5 + x2 * 0.5) = 78
(10x + 0.5x2) = 78

Solving
10x + 0.5x2 = 78

Solving for variable 'x'.

Reorder the terms:
-78 + 10x + 0.5x2 = 78 + -78

Combine like terms: 78 + -78 = 0
-78 + 10x + 0.5x2 = 0

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

Divide each side by '0.5'.
-156 + 20x + x2 = 0

Move the constant term to the right:

Add '156' to each side of the equation.
-156 + 20x + 156 + x2 = 0 + 156

Reorder the terms:
-156 + 156 + 20x + x2 = 0 + 156

Combine like terms: -156 + 156 = 0
0 + 20x + x2 = 0 + 156
20x + x2 = 0 + 156

Combine like terms: 0 + 156 = 156
20x + x2 = 156

The x term is 20x.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20x + 100 + x2 = 156 + 100

Reorder the terms:
100 + 20x + x2 = 156 + 100

Combine like terms: 156 + 100 = 256
100 + 20x + x2 = 256

Factor a perfect square on the left side:
(x + 10)(x + 10) = 256

Calculate the square root of the right side: 16

Break this problem into two subproblems by setting 
(x + 10) equal to 16 and -16.

Subproblem 1

x + 10 = 16 Simplifying x + 10 = 16 Reorder the terms: 10 + x = 16 Solving 10 + x = 16 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = 16 + -10 Combine like terms: 10 + -10 = 0 0 + x = 16 + -10 x = 16 + -10 Combine like terms: 16 + -10 = 6 x = 6 Simplifying x = 6

Subproblem 2

x + 10 = -16 Simplifying x + 10 = -16 Reorder the terms: 10 + x = -16 Solving 10 + x = -16 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = -16 + -10 Combine like terms: 10 + -10 = 0 0 + x = -16 + -10 x = -16 + -10 Combine like terms: -16 + -10 = -26 x = -26 Simplifying x = -26

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6, -26}

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