x(x+3)=84

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Solution for x(x+3)=84 equation: 


Simplifying
x(x + 3) = 84

Reorder the terms:
x(3 + x) = 84
(3 * x + x * x) = 84
(3x + x2) = 84

Solving
3x + x2 = 84

Solving for variable 'x'.

Reorder the terms:
-84 + 3x + x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + 3x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + 3x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + 3x + x2 = 0 + 84
3x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
3x + x2 = 84

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 84 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 84 + 2.25

Combine like terms: 84 + 2.25 = 86.25
2.25 + 3x + x2 = 86.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 86.25

Calculate the square root of the right side: 9.287087811

Break this problem into two subproblems by setting 
(x + 1.5) equal to 9.287087811 and -9.287087811.


Subproblem 1
x + 1.5 = 9.287087811

Simplifying
x + 1.5 = 9.287087811

Reorder the terms:
1.5 + x = 9.287087811

Solving
1.5 + x = 9.287087811

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 9.287087811 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 9.287087811 + -1.5
x = 9.287087811 + -1.5

Combine like terms: 9.287087811 + -1.5 = 7.787087811
x = 7.787087811

Simplifying
x = 7.787087811


Subproblem 2
x + 1.5 = -9.287087811

Simplifying
x + 1.5 = -9.287087811

Reorder the terms:
1.5 + x = -9.287087811

Solving
1.5 + x = -9.287087811

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -9.287087811 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -9.287087811 + -1.5
x = -9.287087811 + -1.5

Combine like terms: -9.287087811 + -1.5 = -10.787087811
x = -10.787087811

Simplifying
x = -10.787087811


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.787087811, -10.787087811}

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