(4x^2)+9x-14=0

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Solution for (4x^2)+9x-14=0 equation: 


Simplifying
(4x2) + 9x + -14 = 0

Reorder the terms:
-14 + 9x + (4x2) = 0

Solving
-14 + 9x + (4x2) = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-3.5 + 2.25x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 2.25x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 2.25x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 2.25x + x2 = 0 + 3.5
2.25x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
2.25x + x2 = 3.5

The x term is 2.25x.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25x + 1.265625 + x2 = 3.5 + 1.265625

Reorder the terms:
1.265625 + 2.25x + x2 = 3.5 + 1.265625

Combine like terms: 3.5 + 1.265625 = 4.765625
1.265625 + 2.25x + x2 = 4.765625

Factor a perfect square on the left side:
((x) + 1.125)((x) + 1.125) = 4.765625

Calculate the square root of the right side: 2.18303115

Break this problem into two subproblems by setting 
((x) + 1.125) equal to 2.18303115 and -2.18303115.


Subproblem 1
(x) + 1.125 = 2.18303115

Simplifying
(x) + 1.125 = 2.18303115
x + 1.125 = 2.18303115

Reorder the terms:
1.125 + x = 2.18303115

Solving
1.125 + x = 2.18303115

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = 2.18303115 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = 2.18303115 + -1.125
x = 2.18303115 + -1.125

Combine like terms: 2.18303115 + -1.125 = 1.05803115
x = 1.05803115

Simplifying
x = 1.05803115


Subproblem 2
(x) + 1.125 = -2.18303115

Simplifying
(x) + 1.125 = -2.18303115
x + 1.125 = -2.18303115

Reorder the terms:
1.125 + x = -2.18303115

Solving
1.125 + x = -2.18303115

Solving for variable 'x'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + x = -2.18303115 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + x = -2.18303115 + -1.125
x = -2.18303115 + -1.125

Combine like terms: -2.18303115 + -1.125 = -3.30803115
x = -3.30803115

Simplifying
x = -3.30803115


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.05803115, -3.30803115}

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