x^2-21x-78=0

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Solution for x^2-21x-78=0 equation:


Simplifying
x2 + -21x + -78 = 0

Reorder the terms:
-78 + -21x + x2 = 0

Solving
-78 + -21x + 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 + -21x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + -21x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -21x + x2 = 0 + 78
-21x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
-21x + x2 = 78

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = 78 + 110.25

Reorder the terms:
110.25 + -21x + x2 = 78 + 110.25

Combine like terms: 78 + 110.25 = 188.25
110.25 + -21x + x2 = 188.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 188.25

Calculate the square root of the right side: 13.720422734

Break this problem into two subproblems by setting 
(x + -10.5) equal to 13.720422734 and -13.720422734.

Subproblem 1

x + -10.5 = 13.720422734 Simplifying x + -10.5 = 13.720422734 Reorder the terms: -10.5 + x = 13.720422734 Solving -10.5 + x = 13.720422734 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = 13.720422734 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 13.720422734 + 10.5 x = 13.720422734 + 10.5 Combine like terms: 13.720422734 + 10.5 = 24.220422734 x = 24.220422734 Simplifying x = 24.220422734

Subproblem 2

x + -10.5 = -13.720422734 Simplifying x + -10.5 = -13.720422734 Reorder the terms: -10.5 + x = -13.720422734 Solving -10.5 + x = -13.720422734 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = -13.720422734 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -13.720422734 + 10.5 x = -13.720422734 + 10.5 Combine like terms: -13.720422734 + 10.5 = -3.220422734 x = -3.220422734 Simplifying x = -3.220422734

Solution

The solution to the problem is based on the solutions from the subproblems. x = {24.220422734, -3.220422734}

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