(-x^2+6)+(2x^2+3x-5)=0

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Solution for (-x^2+6)+(2x^2+3x-5)=0 equation:


Simplifying
(-1x2 + 6) + (2x2 + 3x + -5) = 0

Reorder the terms:
(6 + -1x2) + (2x2 + 3x + -5) = 0

Remove parenthesis around (6 + -1x2)
6 + -1x2 + (2x2 + 3x + -5) = 0

Reorder the terms:
6 + -1x2 + (-5 + 3x + 2x2) = 0

Remove parenthesis around (-5 + 3x + 2x2)
6 + -1x2 + -5 + 3x + 2x2 = 0

Reorder the terms:
6 + -5 + 3x + -1x2 + 2x2 = 0

Combine like terms: 6 + -5 = 1
1 + 3x + -1x2 + 2x2 = 0

Combine like terms: -1x2 + 2x2 = 1x2
1 + 3x + 1x2 = 0

Solving
1 + 3x + 1x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 3x + x2 = 0 + -1

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

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

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = -1 + 2.25

Reorder the terms:
2.25 + 3x + x2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3x + x2 = 1.25

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + 1.5 = 1.118033989 Simplifying x + 1.5 = 1.118033989 Reorder the terms: 1.5 + x = 1.118033989 Solving 1.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 1.118033989 + -1.5 x = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 x = -0.381966011 Simplifying x = -0.381966011

Subproblem 2

x + 1.5 = -1.118033989 Simplifying x + 1.5 = -1.118033989 Reorder the terms: 1.5 + x = -1.118033989 Solving 1.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -1.118033989 + -1.5 x = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 x = -2.618033989 Simplifying x = -2.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.381966011, -2.618033989}

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