.18(24)+.03x=.06x(24+x)

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Solution for .18(24)+.03x=.06x(24+x) equation:


Simplifying
0.18(24) + 0.03x = 0.06x(24 + x)

Multiply 0.18 * 24
4.32 + 0.03x = 0.06x(24 + x)
4.32 + 0.03x = (24 * 0.06x + x * 0.06x)
4.32 + 0.03x = (1.44x + 0.06x2)

Solving
4.32 + 0.03x = 1.44x + 0.06x2

Solving for variable 'x'.

Combine like terms: 0.03x + -1.44x = -1.41x
4.32 + -1.41x + -0.06x2 = 1.44x + 0.06x2 + -1.44x + -0.06x2

Reorder the terms:
4.32 + -1.41x + -0.06x2 = 1.44x + -1.44x + 0.06x2 + -0.06x2

Combine like terms: 1.44x + -1.44x = 0.00
4.32 + -1.41x + -0.06x2 = 0.00 + 0.06x2 + -0.06x2
4.32 + -1.41x + -0.06x2 = 0.06x2 + -0.06x2

Combine like terms: 0.06x2 + -0.06x2 = 0.00
4.32 + -1.41x + -0.06x2 = 0.00

Begin completing the square.  Divide all terms by
-0.06 the coefficient of the squared term: 

Divide each side by '-0.06'.
-72 + 23.5x + x2 = 0

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + 23.5x + 72 + x2 = 0 + 72

Reorder the terms:
-72 + 72 + 23.5x + x2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + 23.5x + x2 = 0 + 72
23.5x + x2 = 0 + 72

Combine like terms: 0 + 72 = 72
23.5x + x2 = 72

The x term is 23.5x.  Take half its coefficient (11.75).
Square it (138.0625) and add it to both sides.

Add '138.0625' to each side of the equation.
23.5x + 138.0625 + x2 = 72 + 138.0625

Reorder the terms:
138.0625 + 23.5x + x2 = 72 + 138.0625

Combine like terms: 72 + 138.0625 = 210.0625
138.0625 + 23.5x + x2 = 210.0625

Factor a perfect square on the left side:
(x + 11.75)(x + 11.75) = 210.0625

Calculate the square root of the right side: 14.493533041

Break this problem into two subproblems by setting 
(x + 11.75) equal to 14.493533041 and -14.493533041.

Subproblem 1

x + 11.75 = 14.493533041 Simplifying x + 11.75 = 14.493533041 Reorder the terms: 11.75 + x = 14.493533041 Solving 11.75 + x = 14.493533041 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.75' to each side of the equation. 11.75 + -11.75 + x = 14.493533041 + -11.75 Combine like terms: 11.75 + -11.75 = 0.00 0.00 + x = 14.493533041 + -11.75 x = 14.493533041 + -11.75 Combine like terms: 14.493533041 + -11.75 = 2.743533041 x = 2.743533041 Simplifying x = 2.743533041

Subproblem 2

x + 11.75 = -14.493533041 Simplifying x + 11.75 = -14.493533041 Reorder the terms: 11.75 + x = -14.493533041 Solving 11.75 + x = -14.493533041 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-11.75' to each side of the equation. 11.75 + -11.75 + x = -14.493533041 + -11.75 Combine like terms: 11.75 + -11.75 = 0.00 0.00 + x = -14.493533041 + -11.75 x = -14.493533041 + -11.75 Combine like terms: -14.493533041 + -11.75 = -26.243533041 x = -26.243533041 Simplifying x = -26.243533041

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.743533041, -26.243533041}

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