x^2-3x-77=0

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Solution for x^2-3x-77=0 equation:


Simplifying
x2 + -3x + -77 = 0

Reorder the terms:
-77 + -3x + x2 = 0

Solving
-77 + -3x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '77' to each side of the equation.
-77 + -3x + 77 + x2 = 0 + 77

Reorder the terms:
-77 + 77 + -3x + x2 = 0 + 77

Combine like terms: -77 + 77 = 0
0 + -3x + x2 = 0 + 77
-3x + x2 = 0 + 77

Combine like terms: 0 + 77 = 77
-3x + x2 = 77

The x term is -3x.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3x + 2.25 + x2 = 77 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 77 + 2.25

Combine like terms: 77 + 2.25 = 79.25
2.25 + -3x + x2 = 79.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 79.25

Calculate the square root of the right side: 8.902246907

Break this problem into two subproblems by setting 
(x + -1.5) equal to 8.902246907 and -8.902246907.

Subproblem 1

x + -1.5 = 8.902246907 Simplifying x + -1.5 = 8.902246907 Reorder the terms: -1.5 + x = 8.902246907 Solving -1.5 + x = 8.902246907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = 8.902246907 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 8.902246907 + 1.5 x = 8.902246907 + 1.5 Combine like terms: 8.902246907 + 1.5 = 10.402246907 x = 10.402246907 Simplifying x = 10.402246907

Subproblem 2

x + -1.5 = -8.902246907 Simplifying x + -1.5 = -8.902246907 Reorder the terms: -1.5 + x = -8.902246907 Solving -1.5 + x = -8.902246907 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + x = -8.902246907 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -8.902246907 + 1.5 x = -8.902246907 + 1.5 Combine like terms: -8.902246907 + 1.5 = -7.402246907 x = -7.402246907 Simplifying x = -7.402246907

Solution

The solution to the problem is based on the solutions from the subproblems. x = {10.402246907, -7.402246907}

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