625=.5x(25+x)

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Solution for 625=.5x(25+x) equation:


Simplifying
625 = 0.5x(25 + x)
625 = (25 * 0.5x + x * 0.5x)
625 = (12.5x + 0.5x2)

Solving
625 = 12.5x + 0.5x2

Solving for variable 'x'.

Reorder the terms:
625 + -12.5x + -0.5x2 = 12.5x + -12.5x + 0.5x2 + -0.5x2

Combine like terms: 12.5x + -12.5x = 0.0
625 + -12.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2
625 + -12.5x + -0.5x2 = 0.5x2 + -0.5x2

Combine like terms: 0.5x2 + -0.5x2 = 0.0
625 + -12.5x + -0.5x2 = 0.0

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

Divide each side by '-0.5'.
-1250 + 25x + x2 = 0

Move the constant term to the right:

Add '1250' to each side of the equation.
-1250 + 25x + 1250 + x2 = 0 + 1250

Reorder the terms:
-1250 + 1250 + 25x + x2 = 0 + 1250

Combine like terms: -1250 + 1250 = 0
0 + 25x + x2 = 0 + 1250
25x + x2 = 0 + 1250

Combine like terms: 0 + 1250 = 1250
25x + x2 = 1250

The x term is 25x.  Take half its coefficient (12.5).
Square it (156.25) and add it to both sides.

Add '156.25' to each side of the equation.
25x + 156.25 + x2 = 1250 + 156.25

Reorder the terms:
156.25 + 25x + x2 = 1250 + 156.25

Combine like terms: 1250 + 156.25 = 1406.25
156.25 + 25x + x2 = 1406.25

Factor a perfect square on the left side:
(x + 12.5)(x + 12.5) = 1406.25

Calculate the square root of the right side: 37.5

Break this problem into two subproblems by setting 
(x + 12.5) equal to 37.5 and -37.5.

Subproblem 1

x + 12.5 = 37.5 Simplifying x + 12.5 = 37.5 Reorder the terms: 12.5 + x = 37.5 Solving 12.5 + x = 37.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = 37.5 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = 37.5 + -12.5 x = 37.5 + -12.5 Combine like terms: 37.5 + -12.5 = 25 x = 25 Simplifying x = 25

Subproblem 2

x + 12.5 = -37.5 Simplifying x + 12.5 = -37.5 Reorder the terms: 12.5 + x = -37.5 Solving 12.5 + x = -37.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12.5' to each side of the equation. 12.5 + -12.5 + x = -37.5 + -12.5 Combine like terms: 12.5 + -12.5 = 0.0 0.0 + x = -37.5 + -12.5 x = -37.5 + -12.5 Combine like terms: -37.5 + -12.5 = -50 x = -50 Simplifying x = -50

Solution

The solution to the problem is based on the solutions from the subproblems. x = {25, -50}

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