4x^2+6x-84=0

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Solution for 4x^2+6x-84=0 equation: 


Simplifying
4x2 + 6x + -84 = 0

Reorder the terms:
-84 + 6x + 4x2 = 0

Solving
-84 + 6x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 3x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-42 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-42 + 3x + 2x2 = 0

Solving
-42 + 3x + 2x2 = 0

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

Divide each side by '2'.
-21 + 1.5x + x2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 1.5x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + 1.5x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 1.5x + x2 = 0 + 21
1.5x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
1.5x + x2 = 21

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 = 21 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 21 + 0.5625

Combine like terms: 21 + 0.5625 = 21.5625
0.5625 + 1.5x + x2 = 21.5625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 21.5625

Calculate the square root of the right side: 4.643543905

Break this problem into two subproblems by setting 
(x + 0.75) equal to 4.643543905 and -4.643543905.


Subproblem 1
x + 0.75 = 4.643543905

Simplifying
x + 0.75 = 4.643543905

Reorder the terms:
0.75 + x = 4.643543905

Solving
0.75 + x = 4.643543905

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 = 4.643543905 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 4.643543905 + -0.75
x = 4.643543905 + -0.75

Combine like terms: 4.643543905 + -0.75 = 3.893543905
x = 3.893543905

Simplifying
x = 3.893543905


Subproblem 2
x + 0.75 = -4.643543905

Simplifying
x + 0.75 = -4.643543905

Reorder the terms:
0.75 + x = -4.643543905

Solving
0.75 + x = -4.643543905

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 = -4.643543905 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -4.643543905 + -0.75
x = -4.643543905 + -0.75

Combine like terms: -4.643543905 + -0.75 = -5.393543905
x = -5.393543905

Simplifying
x = -5.393543905


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.893543905, -5.393543905}
Solution
x = {3.893543905, -5.393543905}

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