(3x^2+5x-8)+(5x^2-13x-5)=0

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Solution for (3x^2+5x-8)+(5x^2-13x-5)=0 equation:


Simplifying
(3x2 + 5x + -8) + (5x2 + -13x + -5) = 0

Reorder the terms:
(-8 + 5x + 3x2) + (5x2 + -13x + -5) = 0

Remove parenthesis around (-8 + 5x + 3x2)
-8 + 5x + 3x2 + (5x2 + -13x + -5) = 0

Reorder the terms:
-8 + 5x + 3x2 + (-5 + -13x + 5x2) = 0

Remove parenthesis around (-5 + -13x + 5x2)
-8 + 5x + 3x2 + -5 + -13x + 5x2 = 0

Reorder the terms:
-8 + -5 + 5x + -13x + 3x2 + 5x2 = 0

Combine like terms: -8 + -5 = -13
-13 + 5x + -13x + 3x2 + 5x2 = 0

Combine like terms: 5x + -13x = -8x
-13 + -8x + 3x2 + 5x2 = 0

Combine like terms: 3x2 + 5x2 = 8x2
-13 + -8x + 8x2 = 0

Solving
-13 + -8x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
-1.625 + -1x + x2 = 0

Move the constant term to the right:

Add '1.625' to each side of the equation.
-1.625 + -1x + 1.625 + x2 = 0 + 1.625

Reorder the terms:
-1.625 + 1.625 + -1x + x2 = 0 + 1.625

Combine like terms: -1.625 + 1.625 = 0.000
0.000 + -1x + x2 = 0 + 1.625
-1x + x2 = 0 + 1.625

Combine like terms: 0 + 1.625 = 1.625
-1x + x2 = 1.625

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 1.625 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 1.625 + 0.25

Combine like terms: 1.625 + 0.25 = 1.875
0.25 + -1x + x2 = 1.875

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 1.875

Calculate the square root of the right side: 1.369306394

Break this problem into two subproblems by setting 
(x + -0.5) equal to 1.369306394 and -1.369306394.

Subproblem 1

x + -0.5 = 1.369306394 Simplifying x + -0.5 = 1.369306394 Reorder the terms: -0.5 + x = 1.369306394 Solving -0.5 + x = 1.369306394 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 1.369306394 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 1.369306394 + 0.5 x = 1.369306394 + 0.5 Combine like terms: 1.369306394 + 0.5 = 1.869306394 x = 1.869306394 Simplifying x = 1.869306394

Subproblem 2

x + -0.5 = -1.369306394 Simplifying x + -0.5 = -1.369306394 Reorder the terms: -0.5 + x = -1.369306394 Solving -0.5 + x = -1.369306394 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -1.369306394 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -1.369306394 + 0.5 x = -1.369306394 + 0.5 Combine like terms: -1.369306394 + 0.5 = -0.869306394 x = -0.869306394 Simplifying x = -0.869306394

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.869306394, -0.869306394}

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