6x^2+6x=7

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


Simplifying
6x2 + 6x = 7

Reorder the terms:
6x + 6x2 = 7

Solving
6x + 6x2 = 7

Solving for variable 'x'.

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

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

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

Divide each side by '6'.
-1.166666667 + x + x2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + x + 1.166666667 + x2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + x + x2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + x + x2 = 0 + 1.166666667
x + x2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
x + x2 = 1.166666667

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

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 1.166666667 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 1.166666667 + 0.25

Combine like terms: 1.166666667 + 0.25 = 1.416666667
1.25 + x2 = 1.416666667

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.416666667

Calculate the square root of the right side: 1.190238072

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.190238072 and -1.190238072.

Subproblem 1

x + 0.5 = 1.190238072 Simplifying x + 0.5 = 1.190238072 Reorder the terms: 0.5 + x = 1.190238072 Solving 0.5 + x = 1.190238072 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.190238072 + -0.5 x = 1.190238072 + -0.5 Combine like terms: 1.190238072 + -0.5 = 0.690238072 x = 0.690238072 Simplifying x = 0.690238072

Subproblem 2

x + 0.5 = -1.190238072 Simplifying x + 0.5 = -1.190238072 Reorder the terms: 0.5 + x = -1.190238072 Solving 0.5 + x = -1.190238072 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.190238072 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.190238072 + -0.5 x = -1.190238072 + -0.5 Combine like terms: -1.190238072 + -0.5 = -1.690238072 x = -1.690238072 Simplifying x = -1.690238072

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.690238072, -1.690238072}

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