7x=3x^2-1

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Solution for 7x=3x^2-1 equation:


Simplifying
7x = 3x2 + -1

Reorder the terms:
7x = -1 + 3x2

Solving
7x = -1 + 3x2

Solving for variable 'x'.

Reorder the terms:
1 + 7x + -3x2 = -1 + 3x2 + 1 + -3x2

Reorder the terms:
1 + 7x + -3x2 = -1 + 1 + 3x2 + -3x2

Combine like terms: -1 + 1 = 0
1 + 7x + -3x2 = 0 + 3x2 + -3x2
1 + 7x + -3x2 = 3x2 + -3x2

Combine like terms: 3x2 + -3x2 = 0
1 + 7x + -3x2 = 0

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

Divide each side by '-3'.
-0.3333333333 + -2.333333333x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -2.333333333x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -2.333333333x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -2.333333333x + x2 = 0 + 0.3333333333
-2.333333333x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-2.333333333x + x2 = 0.3333333333

The x term is -2.333333333x.  Take half its coefficient (-1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
-2.333333333x + 1.361111112 + x2 = 0.3333333333 + 1.361111112

Reorder the terms:
1.361111112 + -2.333333333x + x2 = 0.3333333333 + 1.361111112

Combine like terms: 0.3333333333 + 1.361111112 = 1.6944444453
1.361111112 + -2.333333333x + x2 = 1.6944444453

Factor a perfect square on the left side:
(x + -1.166666667)(x + -1.166666667) = 1.6944444453

Calculate the square root of the right side: 1.30170828

Break this problem into two subproblems by setting 
(x + -1.166666667) equal to 1.30170828 and -1.30170828.

Subproblem 1

x + -1.166666667 = 1.30170828 Simplifying x + -1.166666667 = 1.30170828 Reorder the terms: -1.166666667 + x = 1.30170828 Solving -1.166666667 + x = 1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = 1.30170828 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = 1.30170828 + 1.166666667 x = 1.30170828 + 1.166666667 Combine like terms: 1.30170828 + 1.166666667 = 2.468374947 x = 2.468374947 Simplifying x = 2.468374947

Subproblem 2

x + -1.166666667 = -1.30170828 Simplifying x + -1.166666667 = -1.30170828 Reorder the terms: -1.166666667 + x = -1.30170828 Solving -1.166666667 + x = -1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.166666667' to each side of the equation. -1.166666667 + 1.166666667 + x = -1.30170828 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + x = -1.30170828 + 1.166666667 x = -1.30170828 + 1.166666667 Combine like terms: -1.30170828 + 1.166666667 = -0.135041613 x = -0.135041613 Simplifying x = -0.135041613

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.468374947, -0.135041613}

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