3x^2+5x-6=-x^2+3x+9

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Solution for 3x^2+5x-6=-x^2+3x+9 equation:


Simplifying
3x2 + 5x + -6 = -1x2 + 3x + 9

Reorder the terms:
-6 + 5x + 3x2 = -1x2 + 3x + 9

Reorder the terms:
-6 + 5x + 3x2 = 9 + 3x + -1x2

Solving
-6 + 5x + 3x2 = 9 + 3x + -1x2

Solving for variable 'x'.

Reorder the terms:
-6 + -9 + 5x + -3x + 3x2 + x2 = 9 + 3x + -1x2 + -9 + -3x + x2

Combine like terms: -6 + -9 = -15
-15 + 5x + -3x + 3x2 + x2 = 9 + 3x + -1x2 + -9 + -3x + x2

Combine like terms: 5x + -3x = 2x
-15 + 2x + 3x2 + x2 = 9 + 3x + -1x2 + -9 + -3x + x2

Combine like terms: 3x2 + x2 = 4x2
-15 + 2x + 4x2 = 9 + 3x + -1x2 + -9 + -3x + x2

Reorder the terms:
-15 + 2x + 4x2 = 9 + -9 + 3x + -3x + -1x2 + x2

Combine like terms: 9 + -9 = 0
-15 + 2x + 4x2 = 0 + 3x + -3x + -1x2 + x2
-15 + 2x + 4x2 = 3x + -3x + -1x2 + x2

Combine like terms: 3x + -3x = 0
-15 + 2x + 4x2 = 0 + -1x2 + x2
-15 + 2x + 4x2 = -1x2 + x2

Combine like terms: -1x2 + x2 = 0
-15 + 2x + 4x2 = 0

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

Divide each side by '4'.
-3.75 + 0.5x + x2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + 0.5x + 3.75 + x2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + 0.5x + x2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + 0.5x + x2 = 0 + 3.75
0.5x + x2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
0.5x + x2 = 3.75

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 3.75 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 3.75 + 0.0625

Combine like terms: 3.75 + 0.0625 = 3.8125
0.0625 + 0.5x + x2 = 3.8125

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 3.8125

Calculate the square root of the right side: 1.952562419

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.952562419 and -1.952562419.

Subproblem 1

x + 0.25 = 1.952562419 Simplifying x + 0.25 = 1.952562419 Reorder the terms: 0.25 + x = 1.952562419 Solving 0.25 + x = 1.952562419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.952562419 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.952562419 + -0.25 x = 1.952562419 + -0.25 Combine like terms: 1.952562419 + -0.25 = 1.702562419 x = 1.702562419 Simplifying x = 1.702562419

Subproblem 2

x + 0.25 = -1.952562419 Simplifying x + 0.25 = -1.952562419 Reorder the terms: 0.25 + x = -1.952562419 Solving 0.25 + x = -1.952562419 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.952562419 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.952562419 + -0.25 x = -1.952562419 + -0.25 Combine like terms: -1.952562419 + -0.25 = -2.202562419 x = -2.202562419 Simplifying x = -2.202562419

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.702562419, -2.202562419}

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