x^2+3x=3(1-x^2)

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Solution for x^2+3x=3(1-x^2) equation:


Simplifying
x2 + 3x = 3(1 + -1x2)

Reorder the terms:
3x + x2 = 3(1 + -1x2)
3x + x2 = (1 * 3 + -1x2 * 3)
3x + x2 = (3 + -3x2)

Solving
3x + x2 = 3 + -3x2

Solving for variable 'x'.

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

Combine like terms: x2 + 3x2 = 4x2
-3 + 3x + 4x2 = 3 + -3x2 + -3 + 3x2

Reorder the terms:
-3 + 3x + 4x2 = 3 + -3 + -3x2 + 3x2

Combine like terms: 3 + -3 = 0
-3 + 3x + 4x2 = 0 + -3x2 + 3x2
-3 + 3x + 4x2 = -3x2 + 3x2

Combine like terms: -3x2 + 3x2 = 0
-3 + 3x + 4x2 = 0

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

Divide each side by '4'.
-0.75 + 0.75x + x2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 0.75x + 0.75 + x2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 0.75x + x2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 0.75x + x2 = 0 + 0.75
0.75x + x2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
0.75x + x2 = 0.75

The x term is 0.75x.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75x + 0.140625 + x2 = 0.75 + 0.140625

Reorder the terms:
0.140625 + 0.75x + x2 = 0.75 + 0.140625

Combine like terms: 0.75 + 0.140625 = 0.890625
0.140625 + 0.75x + x2 = 0.890625

Factor a perfect square on the left side:
(x + 0.375)(x + 0.375) = 0.890625

Calculate the square root of the right side: 0.943729304

Break this problem into two subproblems by setting 
(x + 0.375) equal to 0.943729304 and -0.943729304.

Subproblem 1

x + 0.375 = 0.943729304 Simplifying x + 0.375 = 0.943729304 Reorder the terms: 0.375 + x = 0.943729304 Solving 0.375 + x = 0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 0.943729304 + -0.375 x = 0.943729304 + -0.375 Combine like terms: 0.943729304 + -0.375 = 0.568729304 x = 0.568729304 Simplifying x = 0.568729304

Subproblem 2

x + 0.375 = -0.943729304 Simplifying x + 0.375 = -0.943729304 Reorder the terms: 0.375 + x = -0.943729304 Solving 0.375 + x = -0.943729304 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -0.943729304 + -0.375 x = -0.943729304 + -0.375 Combine like terms: -0.943729304 + -0.375 = -1.318729304 x = -1.318729304 Simplifying x = -1.318729304

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.568729304, -1.318729304}

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