3x^2=6x+7

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Solution for 3x^2=6x+7 equation:


Simplifying
3x2 = 6x + 7

Reorder the terms:
3x2 = 7 + 6x

Solving
3x2 = 7 + 6x

Solving for variable 'x'.

Reorder the terms:
-7 + -6x + 3x2 = 7 + 6x + -7 + -6x

Reorder the terms:
-7 + -6x + 3x2 = 7 + -7 + 6x + -6x

Combine like terms: 7 + -7 = 0
-7 + -6x + 3x2 = 0 + 6x + -6x
-7 + -6x + 3x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-7 + -6x + 3x2 = 0

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

Divide each side by '3'.
-2.333333333 + -2x + x2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + -2x + 2.333333333 + x2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + -2x + x2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + -2x + x2 = 0 + 2.333333333
-2x + x2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
-2x + x2 = 2.333333333

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 2.333333333 + 1

Reorder the terms:
1 + -2x + x2 = 2.333333333 + 1

Combine like terms: 2.333333333 + 1 = 3.333333333
1 + -2x + x2 = 3.333333333

Factor a perfect square on the left side:
(x + -1)(x + -1) = 3.333333333

Calculate the square root of the right side: 1.825741858

Break this problem into two subproblems by setting 
(x + -1) equal to 1.825741858 and -1.825741858.

Subproblem 1

x + -1 = 1.825741858 Simplifying x + -1 = 1.825741858 Reorder the terms: -1 + x = 1.825741858 Solving -1 + x = 1.825741858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 1.825741858 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.825741858 + 1 x = 1.825741858 + 1 Combine like terms: 1.825741858 + 1 = 2.825741858 x = 2.825741858 Simplifying x = 2.825741858

Subproblem 2

x + -1 = -1.825741858 Simplifying x + -1 = -1.825741858 Reorder the terms: -1 + x = -1.825741858 Solving -1 + x = -1.825741858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -1.825741858 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.825741858 + 1 x = -1.825741858 + 1 Combine like terms: -1.825741858 + 1 = -0.825741858 x = -0.825741858 Simplifying x = -0.825741858

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.825741858, -0.825741858}

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