2x^2-41x+78=0

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Solution for 2x^2-41x+78=0 equation:


Simplifying
2x2 + -41x + 78 = 0

Reorder the terms:
78 + -41x + 2x2 = 0

Solving
78 + -41x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
39 + -20.5x + x2 = 0

Move the constant term to the right:

Add '-39' to each side of the equation.
39 + -20.5x + -39 + x2 = 0 + -39

Reorder the terms:
39 + -39 + -20.5x + x2 = 0 + -39

Combine like terms: 39 + -39 = 0
0 + -20.5x + x2 = 0 + -39
-20.5x + x2 = 0 + -39

Combine like terms: 0 + -39 = -39
-20.5x + x2 = -39

The x term is -20.5x.  Take half its coefficient (-10.25).
Square it (105.0625) and add it to both sides.

Add '105.0625' to each side of the equation.
-20.5x + 105.0625 + x2 = -39 + 105.0625

Reorder the terms:
105.0625 + -20.5x + x2 = -39 + 105.0625

Combine like terms: -39 + 105.0625 = 66.0625
105.0625 + -20.5x + x2 = 66.0625

Factor a perfect square on the left side:
(x + -10.25)(x + -10.25) = 66.0625

Calculate the square root of the right side: 8.127884104

Break this problem into two subproblems by setting 
(x + -10.25) equal to 8.127884104 and -8.127884104.

Subproblem 1

x + -10.25 = 8.127884104 Simplifying x + -10.25 = 8.127884104 Reorder the terms: -10.25 + x = 8.127884104 Solving -10.25 + x = 8.127884104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.25' to each side of the equation. -10.25 + 10.25 + x = 8.127884104 + 10.25 Combine like terms: -10.25 + 10.25 = 0.00 0.00 + x = 8.127884104 + 10.25 x = 8.127884104 + 10.25 Combine like terms: 8.127884104 + 10.25 = 18.377884104 x = 18.377884104 Simplifying x = 18.377884104

Subproblem 2

x + -10.25 = -8.127884104 Simplifying x + -10.25 = -8.127884104 Reorder the terms: -10.25 + x = -8.127884104 Solving -10.25 + x = -8.127884104 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.25' to each side of the equation. -10.25 + 10.25 + x = -8.127884104 + 10.25 Combine like terms: -10.25 + 10.25 = 0.00 0.00 + x = -8.127884104 + 10.25 x = -8.127884104 + 10.25 Combine like terms: -8.127884104 + 10.25 = 2.122115896 x = 2.122115896 Simplifying x = 2.122115896

Solution

The solution to the problem is based on the solutions from the subproblems. x = {18.377884104, 2.122115896}

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