6x^2+23x+5=0

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Solution for 6x^2+23x+5=0 equation:


Simplifying
6x2 + 23x + 5 = 0

Reorder the terms:
5 + 23x + 6x2 = 0

Solving
5 + 23x + 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'.
0.8333333333 + 3.833333333x + x2 = 0

Move the constant term to the right:

Add '-0.8333333333' to each side of the equation.
0.8333333333 + 3.833333333x + -0.8333333333 + x2 = 0 + -0.8333333333

Reorder the terms:
0.8333333333 + -0.8333333333 + 3.833333333x + x2 = 0 + -0.8333333333

Combine like terms: 0.8333333333 + -0.8333333333 = 0.0000000000
0.0000000000 + 3.833333333x + x2 = 0 + -0.8333333333
3.833333333x + x2 = 0 + -0.8333333333

Combine like terms: 0 + -0.8333333333 = -0.8333333333
3.833333333x + x2 = -0.8333333333

The x term is 3.833333333x.  Take half its coefficient (1.916666667).
Square it (3.673611112) and add it to both sides.

Add '3.673611112' to each side of the equation.
3.833333333x + 3.673611112 + x2 = -0.8333333333 + 3.673611112

Reorder the terms:
3.673611112 + 3.833333333x + x2 = -0.8333333333 + 3.673611112

Combine like terms: -0.8333333333 + 3.673611112 = 2.8402777787
3.673611112 + 3.833333333x + x2 = 2.8402777787

Factor a perfect square on the left side:
(x + 1.916666667)(x + 1.916666667) = 2.8402777787

Calculate the square root of the right side: 1.685312368

Break this problem into two subproblems by setting 
(x + 1.916666667) equal to 1.685312368 and -1.685312368.

Subproblem 1

x + 1.916666667 = 1.685312368 Simplifying x + 1.916666667 = 1.685312368 Reorder the terms: 1.916666667 + x = 1.685312368 Solving 1.916666667 + x = 1.685312368 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = 1.685312368 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = 1.685312368 + -1.916666667 x = 1.685312368 + -1.916666667 Combine like terms: 1.685312368 + -1.916666667 = -0.231354299 x = -0.231354299 Simplifying x = -0.231354299

Subproblem 2

x + 1.916666667 = -1.685312368 Simplifying x + 1.916666667 = -1.685312368 Reorder the terms: 1.916666667 + x = -1.685312368 Solving 1.916666667 + x = -1.685312368 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = -1.685312368 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = -1.685312368 + -1.916666667 x = -1.685312368 + -1.916666667 Combine like terms: -1.685312368 + -1.916666667 = -3.601979035 x = -3.601979035 Simplifying x = -3.601979035

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.231354299, -3.601979035}

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