0=4x^2-4x-19

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Solution for 0=4x^2-4x-19 equation:


Simplifying
0 = 4x2 + -4x + -19

Reorder the terms:
0 = -19 + -4x + 4x2

Solving
0 = -19 + -4x + 4x2

Solving for variable 'x'.

Combine like terms: 0 + 19 = 19
19 + 4x + -4x2 = -19 + -4x + 4x2 + 19 + 4x + -4x2

Reorder the terms:
19 + 4x + -4x2 = -19 + 19 + -4x + 4x + 4x2 + -4x2

Combine like terms: -19 + 19 = 0
19 + 4x + -4x2 = 0 + -4x + 4x + 4x2 + -4x2
19 + 4x + -4x2 = -4x + 4x + 4x2 + -4x2

Combine like terms: -4x + 4x = 0
19 + 4x + -4x2 = 0 + 4x2 + -4x2
19 + 4x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
19 + 4x + -4x2 = 0

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

Divide each side by '-4'.
-4.75 + -1x + x2 = 0

Move the constant term to the right:

Add '4.75' to each side of the equation.
-4.75 + -1x + 4.75 + x2 = 0 + 4.75

Reorder the terms:
-4.75 + 4.75 + -1x + x2 = 0 + 4.75

Combine like terms: -4.75 + 4.75 = 0.00
0.00 + -1x + x2 = 0 + 4.75
-1x + x2 = 0 + 4.75

Combine like terms: 0 + 4.75 = 4.75
-1x + x2 = 4.75

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 4.75 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 4.75 + 0.25

Combine like terms: 4.75 + 0.25 = 5
0.25 + -1x + x2 = 5

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(x + -0.5) equal to 2.236067978 and -2.236067978.

Subproblem 1

x + -0.5 = 2.236067978 Simplifying x + -0.5 = 2.236067978 Reorder the terms: -0.5 + x = 2.236067978 Solving -0.5 + x = 2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 2.236067978 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 2.236067978 + 0.5 x = 2.236067978 + 0.5 Combine like terms: 2.236067978 + 0.5 = 2.736067978 x = 2.736067978 Simplifying x = 2.736067978

Subproblem 2

x + -0.5 = -2.236067978 Simplifying x + -0.5 = -2.236067978 Reorder the terms: -0.5 + x = -2.236067978 Solving -0.5 + x = -2.236067978 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -2.236067978 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -2.236067978 + 0.5 x = -2.236067978 + 0.5 Combine like terms: -2.236067978 + 0.5 = -1.736067978 x = -1.736067978 Simplifying x = -1.736067978

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.736067978, -1.736067978}

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