125=-.5x^2+25x-150

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Solution for 125=-.5x^2+25x-150 equation:


Simplifying
125 = -0.5x2 + 25x + -150

Reorder the terms:
125 = -150 + 25x + -0.5x2

Solving
125 = -150 + 25x + -0.5x2

Solving for variable 'x'.

Combine like terms: 125 + 150 = 275
275 + -25x + 0.5x2 = -150 + 25x + -0.5x2 + 150 + -25x + 0.5x2

Reorder the terms:
275 + -25x + 0.5x2 = -150 + 150 + 25x + -25x + -0.5x2 + 0.5x2

Combine like terms: -150 + 150 = 0
275 + -25x + 0.5x2 = 0 + 25x + -25x + -0.5x2 + 0.5x2
275 + -25x + 0.5x2 = 25x + -25x + -0.5x2 + 0.5x2

Combine like terms: 25x + -25x = 0
275 + -25x + 0.5x2 = 0 + -0.5x2 + 0.5x2
275 + -25x + 0.5x2 = -0.5x2 + 0.5x2

Combine like terms: -0.5x2 + 0.5x2 = 0.0
275 + -25x + 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'.
550 + -50x + x2 = 0

Move the constant term to the right:

Add '-550' to each side of the equation.
550 + -50x + -550 + x2 = 0 + -550

Reorder the terms:
550 + -550 + -50x + x2 = 0 + -550

Combine like terms: 550 + -550 = 0
0 + -50x + x2 = 0 + -550
-50x + x2 = 0 + -550

Combine like terms: 0 + -550 = -550
-50x + x2 = -550

The x term is -50x.  Take half its coefficient (-25).
Square it (625) and add it to both sides.

Add '625' to each side of the equation.
-50x + 625 + x2 = -550 + 625

Reorder the terms:
625 + -50x + x2 = -550 + 625

Combine like terms: -550 + 625 = 75
625 + -50x + x2 = 75

Factor a perfect square on the left side:
(x + -25)(x + -25) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(x + -25) equal to 8.660254038 and -8.660254038.

Subproblem 1

x + -25 = 8.660254038 Simplifying x + -25 = 8.660254038 Reorder the terms: -25 + x = 8.660254038 Solving -25 + x = 8.660254038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '25' to each side of the equation. -25 + 25 + x = 8.660254038 + 25 Combine like terms: -25 + 25 = 0 0 + x = 8.660254038 + 25 x = 8.660254038 + 25 Combine like terms: 8.660254038 + 25 = 33.660254038 x = 33.660254038 Simplifying x = 33.660254038

Subproblem 2

x + -25 = -8.660254038 Simplifying x + -25 = -8.660254038 Reorder the terms: -25 + x = -8.660254038 Solving -25 + x = -8.660254038 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '25' to each side of the equation. -25 + 25 + x = -8.660254038 + 25 Combine like terms: -25 + 25 = 0 0 + x = -8.660254038 + 25 x = -8.660254038 + 25 Combine like terms: -8.660254038 + 25 = 16.339745962 x = 16.339745962 Simplifying x = 16.339745962

Solution

The solution to the problem is based on the solutions from the subproblems. x = {33.660254038, 16.339745962}

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