.07x+.09x(1500-x)=409

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Solution for .07x+.09x(1500-x)=409 equation:


Simplifying
0.07x + 0.09x(1500 + -1x) = 409
0.07x + (1500 * 0.09x + -1x * 0.09x) = 409
0.07x + (135x + -0.09x2) = 409

Combine like terms: 0.07x + 135x = 135.07x
135.07x + -0.09x2 = 409

Solving
135.07x + -0.09x2 = 409

Solving for variable 'x'.

Reorder the terms:
-409 + 135.07x + -0.09x2 = 409 + -409

Combine like terms: 409 + -409 = 0
-409 + 135.07x + -0.09x2 = 0

Begin completing the square.  Divide all terms by
-0.09 the coefficient of the squared term: 

Divide each side by '-0.09'.
4544.444444 + -1500.777778x + x2 = 0

Move the constant term to the right:

Add '-4544.444444' to each side of the equation.
4544.444444 + -1500.777778x + -4544.444444 + x2 = 0 + -4544.444444

Reorder the terms:
4544.444444 + -4544.444444 + -1500.777778x + x2 = 0 + -4544.444444

Combine like terms: 4544.444444 + -4544.444444 = 0.000000
0.000000 + -1500.777778x + x2 = 0 + -4544.444444
-1500.777778x + x2 = 0 + -4544.444444

Combine like terms: 0 + -4544.444444 = -4544.444444
-1500.777778x + x2 = -4544.444444

The x term is -1500.777778x.  Take half its coefficient (-750.388889).
Square it (563083.4847) and add it to both sides.

Add '563083.4847' to each side of the equation.
-1500.777778x + 563083.4847 + x2 = -4544.444444 + 563083.4847

Reorder the terms:
563083.4847 + -1500.777778x + x2 = -4544.444444 + 563083.4847

Combine like terms: -4544.444444 + 563083.4847 = 558539.040256
563083.4847 + -1500.777778x + x2 = 558539.040256

Factor a perfect square on the left side:
(x + -750.388889)(x + -750.388889) = 558539.040256

Calculate the square root of the right side: 747.354695079

Break this problem into two subproblems by setting 
(x + -750.388889) equal to 747.354695079 and -747.354695079.

Subproblem 1

x + -750.388889 = 747.354695079 Simplifying x + -750.388889 = 747.354695079 Reorder the terms: -750.388889 + x = 747.354695079 Solving -750.388889 + x = 747.354695079 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '750.388889' to each side of the equation. -750.388889 + 750.388889 + x = 747.354695079 + 750.388889 Combine like terms: -750.388889 + 750.388889 = 0.000000 0.000000 + x = 747.354695079 + 750.388889 x = 747.354695079 + 750.388889 Combine like terms: 747.354695079 + 750.388889 = 1497.743584079 x = 1497.743584079 Simplifying x = 1497.743584079

Subproblem 2

x + -750.388889 = -747.354695079 Simplifying x + -750.388889 = -747.354695079 Reorder the terms: -750.388889 + x = -747.354695079 Solving -750.388889 + x = -747.354695079 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '750.388889' to each side of the equation. -750.388889 + 750.388889 + x = -747.354695079 + 750.388889 Combine like terms: -750.388889 + 750.388889 = 0.000000 0.000000 + x = -747.354695079 + 750.388889 x = -747.354695079 + 750.388889 Combine like terms: -747.354695079 + 750.388889 = 3.034193921 x = 3.034193921 Simplifying x = 3.034193921

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1497.743584079, 3.034193921}

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