13x^2+10x-13=0

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Solution for 13x^2+10x-13=0 equation:


Simplifying
13x2 + 10x + -13 = 0

Reorder the terms:
-13 + 10x + 13x2 = 0

Solving
-13 + 10x + 13x2 = 0

Solving for variable 'x'.

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

Divide each side by '13'.
-1 + 0.7692307692x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.7692307692x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.7692307692x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.7692307692x + x2 = 0 + 1
0.7692307692x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.7692307692x + x2 = 1

The x term is 0.7692307692x.  Take half its coefficient (0.3846153846).
Square it (0.1479289941) and add it to both sides.

Add '0.1479289941' to each side of the equation.
0.7692307692x + 0.1479289941 + x2 = 1 + 0.1479289941

Reorder the terms:
0.1479289941 + 0.7692307692x + x2 = 1 + 0.1479289941

Combine like terms: 1 + 0.1479289941 = 1.1479289941
0.1479289941 + 0.7692307692x + x2 = 1.1479289941

Factor a perfect square on the left side:
(x + 0.3846153846)(x + 0.3846153846) = 1.1479289941

Calculate the square root of the right side: 1.071414483

Break this problem into two subproblems by setting 
(x + 0.3846153846) equal to 1.071414483 and -1.071414483.

Subproblem 1

x + 0.3846153846 = 1.071414483 Simplifying x + 0.3846153846 = 1.071414483 Reorder the terms: 0.3846153846 + x = 1.071414483 Solving 0.3846153846 + x = 1.071414483 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3846153846' to each side of the equation. 0.3846153846 + -0.3846153846 + x = 1.071414483 + -0.3846153846 Combine like terms: 0.3846153846 + -0.3846153846 = 0.0000000000 0.0000000000 + x = 1.071414483 + -0.3846153846 x = 1.071414483 + -0.3846153846 Combine like terms: 1.071414483 + -0.3846153846 = 0.6867990984 x = 0.6867990984 Simplifying x = 0.6867990984

Subproblem 2

x + 0.3846153846 = -1.071414483 Simplifying x + 0.3846153846 = -1.071414483 Reorder the terms: 0.3846153846 + x = -1.071414483 Solving 0.3846153846 + x = -1.071414483 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3846153846' to each side of the equation. 0.3846153846 + -0.3846153846 + x = -1.071414483 + -0.3846153846 Combine like terms: 0.3846153846 + -0.3846153846 = 0.0000000000 0.0000000000 + x = -1.071414483 + -0.3846153846 x = -1.071414483 + -0.3846153846 Combine like terms: -1.071414483 + -0.3846153846 = -1.4560298676 x = -1.4560298676 Simplifying x = -1.4560298676

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.6867990984, -1.4560298676}

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