x^2-3x-106=0

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Solution for x^2-3x-106=0 equation:


Simplifying
x2 + -3x + -106 = 0

Reorder the terms:
-106 + -3x + x2 = 0

Solving
-106 + -3x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '106' to each side of the equation.
-106 + -3x + 106 + x2 = 0 + 106

Reorder the terms:
-106 + 106 + -3x + x2 = 0 + 106

Combine like terms: -106 + 106 = 0
0 + -3x + x2 = 0 + 106
-3x + x2 = 0 + 106

Combine like terms: 0 + 106 = 106
-3x + x2 = 106

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 = 106 + 2.25

Reorder the terms:
2.25 + -3x + x2 = 106 + 2.25

Combine like terms: 106 + 2.25 = 108.25
2.25 + -3x + x2 = 108.25

Factor a perfect square on the left side:
(x + -1.5)(x + -1.5) = 108.25

Calculate the square root of the right side: 10.404326023

Break this problem into two subproblems by setting 
(x + -1.5) equal to 10.404326023 and -10.404326023.

Subproblem 1

x + -1.5 = 10.404326023 Simplifying x + -1.5 = 10.404326023 Reorder the terms: -1.5 + x = 10.404326023 Solving -1.5 + x = 10.404326023 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 = 10.404326023 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = 10.404326023 + 1.5 x = 10.404326023 + 1.5 Combine like terms: 10.404326023 + 1.5 = 11.904326023 x = 11.904326023 Simplifying x = 11.904326023

Subproblem 2

x + -1.5 = -10.404326023 Simplifying x + -1.5 = -10.404326023 Reorder the terms: -1.5 + x = -10.404326023 Solving -1.5 + x = -10.404326023 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 = -10.404326023 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + x = -10.404326023 + 1.5 x = -10.404326023 + 1.5 Combine like terms: -10.404326023 + 1.5 = -8.904326023 x = -8.904326023 Simplifying x = -8.904326023

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.904326023, -8.904326023}

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