5x^2-3x-6=0

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Solution for 5x^2-3x-6=0 equation:


Simplifying
5x2 + -3x + -6 = 0

Reorder the terms:
-6 + -3x + 5x2 = 0

Solving
-6 + -3x + 5x2 = 0

Solving for variable 'x'.

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

Divide each side by '5'.
-1.2 + -0.6x + x2 = 0

Move the constant term to the right:

Add '1.2' to each side of the equation.
-1.2 + -0.6x + 1.2 + x2 = 0 + 1.2

Reorder the terms:
-1.2 + 1.2 + -0.6x + x2 = 0 + 1.2

Combine like terms: -1.2 + 1.2 = 0.0
0.0 + -0.6x + x2 = 0 + 1.2
-0.6x + x2 = 0 + 1.2

Combine like terms: 0 + 1.2 = 1.2
-0.6x + x2 = 1.2

The x term is -0.6x.  Take half its coefficient (-0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
-0.6x + 0.09 + x2 = 1.2 + 0.09

Reorder the terms:
0.09 + -0.6x + x2 = 1.2 + 0.09

Combine like terms: 1.2 + 0.09 = 1.29
0.09 + -0.6x + x2 = 1.29

Factor a perfect square on the left side:
(x + -0.3)(x + -0.3) = 1.29

Calculate the square root of the right side: 1.135781669

Break this problem into two subproblems by setting 
(x + -0.3) equal to 1.135781669 and -1.135781669.

Subproblem 1

x + -0.3 = 1.135781669 Simplifying x + -0.3 = 1.135781669 Reorder the terms: -0.3 + x = 1.135781669 Solving -0.3 + x = 1.135781669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = 1.135781669 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = 1.135781669 + 0.3 x = 1.135781669 + 0.3 Combine like terms: 1.135781669 + 0.3 = 1.435781669 x = 1.435781669 Simplifying x = 1.435781669

Subproblem 2

x + -0.3 = -1.135781669 Simplifying x + -0.3 = -1.135781669 Reorder the terms: -0.3 + x = -1.135781669 Solving -0.3 + x = -1.135781669 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3' to each side of the equation. -0.3 + 0.3 + x = -1.135781669 + 0.3 Combine like terms: -0.3 + 0.3 = 0.0 0.0 + x = -1.135781669 + 0.3 x = -1.135781669 + 0.3 Combine like terms: -1.135781669 + 0.3 = -0.835781669 x = -0.835781669 Simplifying x = -0.835781669

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.435781669, -0.835781669}

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