x(2x+3)=216

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Solution for x(2x+3)=216 equation:


Simplifying
x(2x + 3) = 216

Reorder the terms:
x(3 + 2x) = 216
(3 * x + 2x * x) = 216
(3x + 2x2) = 216

Solving
3x + 2x2 = 216

Solving for variable 'x'.

Reorder the terms:
-216 + 3x + 2x2 = 216 + -216

Combine like terms: 216 + -216 = 0
-216 + 3x + 2x2 = 0

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

Divide each side by '2'.
-108 + 1.5x + x2 = 0

Move the constant term to the right:

Add '108' to each side of the equation.
-108 + 1.5x + 108 + x2 = 0 + 108

Reorder the terms:
-108 + 108 + 1.5x + x2 = 0 + 108

Combine like terms: -108 + 108 = 0
0 + 1.5x + x2 = 0 + 108
1.5x + x2 = 0 + 108

Combine like terms: 0 + 108 = 108
1.5x + x2 = 108

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 108 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 108 + 0.5625

Combine like terms: 108 + 0.5625 = 108.5625
0.5625 + 1.5x + x2 = 108.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 108.5625

Calculate the square root of the right side: 10.419332992

Break this problem into two subproblems by setting 
(x + 0.75) equal to 10.419332992 and -10.419332992.

Subproblem 1

x + 0.75 = 10.419332992 Simplifying x + 0.75 = 10.419332992 Reorder the terms: 0.75 + x = 10.419332992 Solving 0.75 + x = 10.419332992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 10.419332992 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 10.419332992 + -0.75 x = 10.419332992 + -0.75 Combine like terms: 10.419332992 + -0.75 = 9.669332992 x = 9.669332992 Simplifying x = 9.669332992

Subproblem 2

x + 0.75 = -10.419332992 Simplifying x + 0.75 = -10.419332992 Reorder the terms: 0.75 + x = -10.419332992 Solving 0.75 + x = -10.419332992 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -10.419332992 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -10.419332992 + -0.75 x = -10.419332992 + -0.75 Combine like terms: -10.419332992 + -0.75 = -11.169332992 x = -11.169332992 Simplifying x = -11.169332992

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.669332992, -11.169332992}

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