4x^2-4x-65=0

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Solution for 4x^2-4x-65=0 equation:


Simplifying
4x2 + -4x + -65 = 0

Reorder the terms:
-65 + -4x + 4x2 = 0

Solving
-65 + -4x + 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'.
-16.25 + -1x + x2 = 0

Move the constant term to the right:

Add '16.25' to each side of the equation.
-16.25 + -1x + 16.25 + x2 = 0 + 16.25

Reorder the terms:
-16.25 + 16.25 + -1x + x2 = 0 + 16.25

Combine like terms: -16.25 + 16.25 = 0.00
0.00 + -1x + x2 = 0 + 16.25
-1x + x2 = 0 + 16.25

Combine like terms: 0 + 16.25 = 16.25
-1x + x2 = 16.25

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 = 16.25 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 16.25 + 0.25

Combine like terms: 16.25 + 0.25 = 16.5
0.25 + -1x + x2 = 16.5

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 16.5

Calculate the square root of the right side: 4.062019202

Break this problem into two subproblems by setting 
(x + -0.5) equal to 4.062019202 and -4.062019202.

Subproblem 1

x + -0.5 = 4.062019202 Simplifying x + -0.5 = 4.062019202 Reorder the terms: -0.5 + x = 4.062019202 Solving -0.5 + x = 4.062019202 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 = 4.062019202 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 4.062019202 + 0.5 x = 4.062019202 + 0.5 Combine like terms: 4.062019202 + 0.5 = 4.562019202 x = 4.562019202 Simplifying x = 4.562019202

Subproblem 2

x + -0.5 = -4.062019202 Simplifying x + -0.5 = -4.062019202 Reorder the terms: -0.5 + x = -4.062019202 Solving -0.5 + x = -4.062019202 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 = -4.062019202 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -4.062019202 + 0.5 x = -4.062019202 + 0.5 Combine like terms: -4.062019202 + 0.5 = -3.562019202 x = -3.562019202 Simplifying x = -3.562019202

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.562019202, -3.562019202}

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