40+x=2x(9+x)

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Solution for 40+x=2x(9+x) equation:


Simplifying
40 + x = 2x(9 + x)
40 + x = (9 * 2x + x * 2x)
40 + x = (18x + 2x2)

Solving
40 + x = 18x + 2x2

Solving for variable 'x'.

Combine like terms: x + -18x = -17x
40 + -17x + -2x2 = 18x + 2x2 + -18x + -2x2

Reorder the terms:
40 + -17x + -2x2 = 18x + -18x + 2x2 + -2x2

Combine like terms: 18x + -18x = 0
40 + -17x + -2x2 = 0 + 2x2 + -2x2
40 + -17x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
40 + -17x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-20 + 8.5x + x2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 8.5x + 20 + x2 = 0 + 20

Reorder the terms:
-20 + 20 + 8.5x + x2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 8.5x + x2 = 0 + 20
8.5x + x2 = 0 + 20

Combine like terms: 0 + 20 = 20
8.5x + x2 = 20

The x term is 8.5x.  Take half its coefficient (4.25).
Square it (18.0625) and add it to both sides.

Add '18.0625' to each side of the equation.
8.5x + 18.0625 + x2 = 20 + 18.0625

Reorder the terms:
18.0625 + 8.5x + x2 = 20 + 18.0625

Combine like terms: 20 + 18.0625 = 38.0625
18.0625 + 8.5x + x2 = 38.0625

Factor a perfect square on the left side:
(x + 4.25)(x + 4.25) = 38.0625

Calculate the square root of the right side: 6.16948134

Break this problem into two subproblems by setting 
(x + 4.25) equal to 6.16948134 and -6.16948134.

Subproblem 1

x + 4.25 = 6.16948134 Simplifying x + 4.25 = 6.16948134 Reorder the terms: 4.25 + x = 6.16948134 Solving 4.25 + x = 6.16948134 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = 6.16948134 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = 6.16948134 + -4.25 x = 6.16948134 + -4.25 Combine like terms: 6.16948134 + -4.25 = 1.91948134 x = 1.91948134 Simplifying x = 1.91948134

Subproblem 2

x + 4.25 = -6.16948134 Simplifying x + 4.25 = -6.16948134 Reorder the terms: 4.25 + x = -6.16948134 Solving 4.25 + x = -6.16948134 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.25' to each side of the equation. 4.25 + -4.25 + x = -6.16948134 + -4.25 Combine like terms: 4.25 + -4.25 = 0.00 0.00 + x = -6.16948134 + -4.25 x = -6.16948134 + -4.25 Combine like terms: -6.16948134 + -4.25 = -10.41948134 x = -10.41948134 Simplifying x = -10.41948134

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.91948134, -10.41948134}

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