84(x^2-3x-6)=84

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Solution for 84(x^2-3x-6)=84 equation: 


Simplifying
84(x2 + -3x + -6) = 84

Reorder the terms:
84(-6 + -3x + x2) = 84
(-6 * 84 + -3x * 84 + x2 * 84) = 84
(-504 + -252x + 84x2) = 84

Solving
-504 + -252x + 84x2 = 84

Solving for variable 'x'.

Reorder the terms:
-504 + -84 + -252x + 84x2 = 84 + -84

Combine like terms: -504 + -84 = -588
-588 + -252x + 84x2 = 84 + -84

Combine like terms: 84 + -84 = 0
-588 + -252x + 84x2 = 0

Factor out the Greatest Common Factor (GCF), '84'.
84(-7 + -3x + x2) = 0

Ignore the factor 84.

Subproblem 1
Set the factor '(-7 + -3x + x2)' equal to zero and attempt to solve:

Simplifying
-7 + -3x + x2 = 0

Solving
-7 + -3x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -3x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -3x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -3x + x2 = 0 + 7
-3x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-3x + x2 = 7

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 = 7 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 7 + 2.25

Combine like terms: 7 + 2.25 = 9.25
2.25 + -3x + x2 = 9.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 9.25

Calculate the square root of the right side: 3.041381265

Break this problem into two subproblems by setting 
(x + -1.5) equal to 3.041381265 and -3.041381265.


Subproblem 1
x + -1.5 = 3.041381265

Simplifying
x + -1.5 = 3.041381265

Reorder the terms:
-1.5 + x = 3.041381265

Solving
-1.5 + x = 3.041381265

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 = 3.041381265 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = 3.041381265 + 1.5
x = 3.041381265 + 1.5

Combine like terms: 3.041381265 + 1.5 = 4.541381265
x = 4.541381265

Simplifying
x = 4.541381265


Subproblem 2
x + -1.5 = -3.041381265

Simplifying
x + -1.5 = -3.041381265

Reorder the terms:
-1.5 + x = -3.041381265

Solving
-1.5 + x = -3.041381265

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 = -3.041381265 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + x = -3.041381265 + 1.5
x = -3.041381265 + 1.5

Combine like terms: -3.041381265 + 1.5 = -1.541381265
x = -1.541381265

Simplifying
x = -1.541381265


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.541381265, -1.541381265}
Solution
x = {4.541381265, -1.541381265}

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