x(x+7)=84

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Solution for x(x+7)=84 equation: 


Simplifying
x(x + 7) = 84

Reorder the terms:
x(7 + x) = 84
(7 * x + x * x) = 84
(7x + x2) = 84

Solving
7x + x2 = 84

Solving for variable 'x'.

Reorder the terms:
-84 + 7x + x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '84' to each side of the equation.
-84 + 7x + 84 + x2 = 0 + 84

Reorder the terms:
-84 + 84 + 7x + x2 = 0 + 84

Combine like terms: -84 + 84 = 0
0 + 7x + x2 = 0 + 84
7x + x2 = 0 + 84

Combine like terms: 0 + 84 = 84
7x + x2 = 84

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = 84 + 12.25

Reorder the terms:
12.25 + 7x + x2 = 84 + 12.25

Combine like terms: 84 + 12.25 = 96.25
12.25 + 7x + x2 = 96.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 96.25

Calculate the square root of the right side: 9.810708435

Break this problem into two subproblems by setting 
(x + 3.5) equal to 9.810708435 and -9.810708435.


Subproblem 1
x + 3.5 = 9.810708435

Simplifying
x + 3.5 = 9.810708435

Reorder the terms:
3.5 + x = 9.810708435

Solving
3.5 + x = 9.810708435

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 9.810708435 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 9.810708435 + -3.5
x = 9.810708435 + -3.5

Combine like terms: 9.810708435 + -3.5 = 6.310708435
x = 6.310708435

Simplifying
x = 6.310708435


Subproblem 2
x + 3.5 = -9.810708435

Simplifying
x + 3.5 = -9.810708435

Reorder the terms:
3.5 + x = -9.810708435

Solving
3.5 + x = -9.810708435

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -9.810708435 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -9.810708435 + -3.5
x = -9.810708435 + -3.5

Combine like terms: -9.810708435 + -3.5 = -13.310708435
x = -13.310708435

Simplifying
x = -13.310708435


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.310708435, -13.310708435}

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