.45(24)+.05x=.15x(18+x)

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Solution for .45(24)+.05x=.15x(18+x) equation:


Simplifying
0.45(24) + 0.05x = 0.15x(18 + x)

Multiply 0.45 * 24
10.8 + 0.05x = 0.15x(18 + x)
10.8 + 0.05x = (18 * 0.15x + x * 0.15x)
10.8 + 0.05x = (2.7x + 0.15x2)

Solving
10.8 + 0.05x = 2.7x + 0.15x2

Solving for variable 'x'.

Combine like terms: 0.05x + -2.7x = -2.65x
10.8 + -2.65x + -0.15x2 = 2.7x + 0.15x2 + -2.7x + -0.15x2

Reorder the terms:
10.8 + -2.65x + -0.15x2 = 2.7x + -2.7x + 0.15x2 + -0.15x2

Combine like terms: 2.7x + -2.7x = 0.0
10.8 + -2.65x + -0.15x2 = 0.0 + 0.15x2 + -0.15x2
10.8 + -2.65x + -0.15x2 = 0.15x2 + -0.15x2

Combine like terms: 0.15x2 + -0.15x2 = 0.00
10.8 + -2.65x + -0.15x2 = 0.00

Begin completing the square.  Divide all terms by
-0.15 the coefficient of the squared term: 

Divide each side by '-0.15'.
-72 + 17.66666667x + x2 = 0

Move the constant term to the right:

Add '72' to each side of the equation.
-72 + 17.66666667x + 72 + x2 = 0 + 72

Reorder the terms:
-72 + 72 + 17.66666667x + x2 = 0 + 72

Combine like terms: -72 + 72 = 0
0 + 17.66666667x + x2 = 0 + 72
17.66666667x + x2 = 0 + 72

Combine like terms: 0 + 72 = 72
17.66666667x + x2 = 72

The x term is 17.66666667x.  Take half its coefficient (8.833333335).
Square it (78.02777781) and add it to both sides.

Add '78.02777781' to each side of the equation.
17.66666667x + 78.02777781 + x2 = 72 + 78.02777781

Reorder the terms:
78.02777781 + 17.66666667x + x2 = 72 + 78.02777781

Combine like terms: 72 + 78.02777781 = 150.02777781
78.02777781 + 17.66666667x + x2 = 150.02777781

Factor a perfect square on the left side:
(x + 8.833333335)(x + 8.833333335) = 150.02777781

Calculate the square root of the right side: 12.248582686

Break this problem into two subproblems by setting 
(x + 8.833333335) equal to 12.248582686 and -12.248582686.

Subproblem 1

x + 8.833333335 = 12.248582686 Simplifying x + 8.833333335 = 12.248582686 Reorder the terms: 8.833333335 + x = 12.248582686 Solving 8.833333335 + x = 12.248582686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.833333335' to each side of the equation. 8.833333335 + -8.833333335 + x = 12.248582686 + -8.833333335 Combine like terms: 8.833333335 + -8.833333335 = 0.000000000 0.000000000 + x = 12.248582686 + -8.833333335 x = 12.248582686 + -8.833333335 Combine like terms: 12.248582686 + -8.833333335 = 3.415249351 x = 3.415249351 Simplifying x = 3.415249351

Subproblem 2

x + 8.833333335 = -12.248582686 Simplifying x + 8.833333335 = -12.248582686 Reorder the terms: 8.833333335 + x = -12.248582686 Solving 8.833333335 + x = -12.248582686 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.833333335' to each side of the equation. 8.833333335 + -8.833333335 + x = -12.248582686 + -8.833333335 Combine like terms: 8.833333335 + -8.833333335 = 0.000000000 0.000000000 + x = -12.248582686 + -8.833333335 x = -12.248582686 + -8.833333335 Combine like terms: -12.248582686 + -8.833333335 = -21.081916021 x = -21.081916021 Simplifying x = -21.081916021

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.415249351, -21.081916021}

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