3x+5=2x^2-8

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Solution for 3x+5=2x^2-8 equation:


Simplifying
3x + 5 = 2x2 + -8

Reorder the terms:
5 + 3x = 2x2 + -8

Reorder the terms:
5 + 3x = -8 + 2x2

Solving
5 + 3x = -8 + 2x2

Solving for variable 'x'.

Reorder the terms:
5 + 8 + 3x + -2x2 = -8 + 2x2 + 8 + -2x2

Combine like terms: 5 + 8 = 13
13 + 3x + -2x2 = -8 + 2x2 + 8 + -2x2

Reorder the terms:
13 + 3x + -2x2 = -8 + 8 + 2x2 + -2x2

Combine like terms: -8 + 8 = 0
13 + 3x + -2x2 = 0 + 2x2 + -2x2
13 + 3x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
13 + 3x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-6.5 + -1.5x + x2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + -1.5x + 6.5 + x2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + -1.5x + x2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + -1.5x + x2 = 0 + 6.5
-1.5x + x2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
-1.5x + x2 = 6.5

The x term is -1.5x.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5x + 0.5625 + x2 = 6.5 + 0.5625

Reorder the terms:
0.5625 + -1.5x + x2 = 6.5 + 0.5625

Combine like terms: 6.5 + 0.5625 = 7.0625
0.5625 + -1.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + -0.75)(x + -0.75) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + -0.75) equal to 2.657536453 and -2.657536453.

Subproblem 1

x + -0.75 = 2.657536453 Simplifying x + -0.75 = 2.657536453 Reorder the terms: -0.75 + x = 2.657536453 Solving -0.75 + x = 2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 2.657536453 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 2.657536453 + 0.75 x = 2.657536453 + 0.75 Combine like terms: 2.657536453 + 0.75 = 3.407536453 x = 3.407536453 Simplifying x = 3.407536453

Subproblem 2

x + -0.75 = -2.657536453 Simplifying x + -0.75 = -2.657536453 Reorder the terms: -0.75 + x = -2.657536453 Solving -0.75 + x = -2.657536453 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -2.657536453 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -2.657536453 + 0.75 x = -2.657536453 + 0.75 Combine like terms: -2.657536453 + 0.75 = -1.907536453 x = -1.907536453 Simplifying x = -1.907536453

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.407536453, -1.907536453}

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