6x^2+20x+15=0

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Solution for 6x^2+20x+15=0 equation:


Simplifying
6x2 + 20x + 15 = 0

Reorder the terms:
15 + 20x + 6x2 = 0

Solving
15 + 20x + 6x2 = 0

Solving for variable 'x'.

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

Divide each side by '6'.
2.5 + 3.333333333x + x2 = 0

Move the constant term to the right:

Add '-2.5' to each side of the equation.
2.5 + 3.333333333x + -2.5 + x2 = 0 + -2.5

Reorder the terms:
2.5 + -2.5 + 3.333333333x + x2 = 0 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + 3.333333333x + x2 = 0 + -2.5
3.333333333x + x2 = 0 + -2.5

Combine like terms: 0 + -2.5 = -2.5
3.333333333x + x2 = -2.5

The x term is 3.333333333x.  Take half its coefficient (1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
3.333333333x + 2.777777779 + x2 = -2.5 + 2.777777779

Reorder the terms:
2.777777779 + 3.333333333x + x2 = -2.5 + 2.777777779

Combine like terms: -2.5 + 2.777777779 = 0.277777779
2.777777779 + 3.333333333x + x2 = 0.277777779

Factor a perfect square on the left side:
(x + 1.666666667)(x + 1.666666667) = 0.277777779

Calculate the square root of the right side: 0.527046278

Break this problem into two subproblems by setting 
(x + 1.666666667) equal to 0.527046278 and -0.527046278.

Subproblem 1

x + 1.666666667 = 0.527046278 Simplifying x + 1.666666667 = 0.527046278 Reorder the terms: 1.666666667 + x = 0.527046278 Solving 1.666666667 + x = 0.527046278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 0.527046278 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 0.527046278 + -1.666666667 x = 0.527046278 + -1.666666667 Combine like terms: 0.527046278 + -1.666666667 = -1.139620389 x = -1.139620389 Simplifying x = -1.139620389

Subproblem 2

x + 1.666666667 = -0.527046278 Simplifying x + 1.666666667 = -0.527046278 Reorder the terms: 1.666666667 + x = -0.527046278 Solving 1.666666667 + x = -0.527046278 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -0.527046278 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -0.527046278 + -1.666666667 x = -0.527046278 + -1.666666667 Combine like terms: -0.527046278 + -1.666666667 = -2.193712945 x = -2.193712945 Simplifying x = -2.193712945

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-1.139620389, -2.193712945}

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