6x^2+3x=5

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Solution for 6x^2+3x=5 equation:


Simplifying
6x2 + 3x = 5

Reorder the terms:
3x + 6x2 = 5

Solving
3x + 6x2 = 5

Solving for variable 'x'.

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

Combine like terms: 5 + -5 = 0
-5 + 3x + 6x2 = 0

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

Divide each side by '6'.
-0.8333333333 + 0.5x + x2 = 0

Move the constant term to the right:

Add '0.8333333333' to each side of the equation.
-0.8333333333 + 0.5x + 0.8333333333 + x2 = 0 + 0.8333333333

Reorder the terms:
-0.8333333333 + 0.8333333333 + 0.5x + x2 = 0 + 0.8333333333

Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000
0.0000000000 + 0.5x + x2 = 0 + 0.8333333333
0.5x + x2 = 0 + 0.8333333333

Combine like terms: 0 + 0.8333333333 = 0.8333333333
0.5x + x2 = 0.8333333333

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 0.8333333333 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 0.8333333333 + 0.0625

Combine like terms: 0.8333333333 + 0.0625 = 0.8958333333
0.0625 + 0.5x + x2 = 0.8958333333

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 0.8958333333

Calculate the square root of the right side: 0.946484724

Break this problem into two subproblems by setting 
(x + 0.25) equal to 0.946484724 and -0.946484724.

Subproblem 1

x + 0.25 = 0.946484724 Simplifying x + 0.25 = 0.946484724 Reorder the terms: 0.25 + x = 0.946484724 Solving 0.25 + x = 0.946484724 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.946484724 + -0.25 x = 0.946484724 + -0.25 Combine like terms: 0.946484724 + -0.25 = 0.696484724 x = 0.696484724 Simplifying x = 0.696484724

Subproblem 2

x + 0.25 = -0.946484724 Simplifying x + 0.25 = -0.946484724 Reorder the terms: 0.25 + x = -0.946484724 Solving 0.25 + x = -0.946484724 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.946484724 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.946484724 + -0.25 x = -0.946484724 + -0.25 Combine like terms: -0.946484724 + -0.25 = -1.196484724 x = -1.196484724 Simplifying x = -1.196484724

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.696484724, -1.196484724}

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