x^2-40x-78=0

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Solution for x^2-40x-78=0 equation:


Simplifying
x2 + -40x + -78 = 0

Reorder the terms:
-78 + -40x + x2 = 0

Solving
-78 + -40x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + -40x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + -40x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -40x + x2 = 0 + 78
-40x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
-40x + x2 = 78

The x term is -40x.  Take half its coefficient (-20).
Square it (400) and add it to both sides.

Add '400' to each side of the equation.
-40x + 400 + x2 = 78 + 400

Reorder the terms:
400 + -40x + x2 = 78 + 400

Combine like terms: 78 + 400 = 478
400 + -40x + x2 = 478

Factor a perfect square on the left side:
(x + -20)(x + -20) = 478

Calculate the square root of the right side: 21.863211109

Break this problem into two subproblems by setting 
(x + -20) equal to 21.863211109 and -21.863211109.

Subproblem 1

x + -20 = 21.863211109 Simplifying x + -20 = 21.863211109 Reorder the terms: -20 + x = 21.863211109 Solving -20 + x = 21.863211109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + x = 21.863211109 + 20 Combine like terms: -20 + 20 = 0 0 + x = 21.863211109 + 20 x = 21.863211109 + 20 Combine like terms: 21.863211109 + 20 = 41.863211109 x = 41.863211109 Simplifying x = 41.863211109

Subproblem 2

x + -20 = -21.863211109 Simplifying x + -20 = -21.863211109 Reorder the terms: -20 + x = -21.863211109 Solving -20 + x = -21.863211109 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + x = -21.863211109 + 20 Combine like terms: -20 + 20 = 0 0 + x = -21.863211109 + 20 x = -21.863211109 + 20 Combine like terms: -21.863211109 + 20 = -1.863211109 x = -1.863211109 Simplifying x = -1.863211109

Solution

The solution to the problem is based on the solutions from the subproblems. x = {41.863211109, -1.863211109}

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