6x^2=3x+5

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


Simplifying
6x2 = 3x + 5

Reorder the terms:
6x2 = 5 + 3x

Solving
6x2 = 5 + 3x

Solving for variable 'x'.

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

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

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

Combine like terms: 3x + -3x = 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 = 1.196484724 x = 1.196484724 Simplifying x = 1.196484724

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 = -0.696484724 x = -0.696484724 Simplifying x = -0.696484724

Solution

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

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