13x^2+6x=20

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Solution for 13x^2+6x=20 equation:


Simplifying
13x2 + 6x = 20

Reorder the terms:
6x + 13x2 = 20

Solving
6x + 13x2 = 20

Solving for variable 'x'.

Reorder the terms:
-20 + 6x + 13x2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + 6x + 13x2 = 0

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

Divide each side by '13'.
-1.538461538 + 0.4615384615x + x2 = 0

Move the constant term to the right:

Add '1.538461538' to each side of the equation.
-1.538461538 + 0.4615384615x + 1.538461538 + x2 = 0 + 1.538461538

Reorder the terms:
-1.538461538 + 1.538461538 + 0.4615384615x + x2 = 0 + 1.538461538

Combine like terms: -1.538461538 + 1.538461538 = 0.000000000
0.000000000 + 0.4615384615x + x2 = 0 + 1.538461538
0.4615384615x + x2 = 0 + 1.538461538

Combine like terms: 0 + 1.538461538 = 1.538461538
0.4615384615x + x2 = 1.538461538

The x term is 0.4615384615x.  Take half its coefficient (0.2307692308).
Square it (0.05325443788) and add it to both sides.

Add '0.05325443788' to each side of the equation.
0.4615384615x + 0.05325443788 + x2 = 1.538461538 + 0.05325443788

Reorder the terms:
0.05325443788 + 0.4615384615x + x2 = 1.538461538 + 0.05325443788

Combine like terms: 1.538461538 + 0.05325443788 = 1.59171597588
0.05325443788 + 0.4615384615x + x2 = 1.59171597588

Factor a perfect square on the left side:
(x + 0.2307692308)(x + 0.2307692308) = 1.59171597588

Calculate the square root of the right side: 1.261632267

Break this problem into two subproblems by setting 
(x + 0.2307692308) equal to 1.261632267 and -1.261632267.

Subproblem 1

x + 0.2307692308 = 1.261632267 Simplifying x + 0.2307692308 = 1.261632267 Reorder the terms: 0.2307692308 + x = 1.261632267 Solving 0.2307692308 + x = 1.261632267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2307692308' to each side of the equation. 0.2307692308 + -0.2307692308 + x = 1.261632267 + -0.2307692308 Combine like terms: 0.2307692308 + -0.2307692308 = 0.0000000000 0.0000000000 + x = 1.261632267 + -0.2307692308 x = 1.261632267 + -0.2307692308 Combine like terms: 1.261632267 + -0.2307692308 = 1.0308630362 x = 1.0308630362 Simplifying x = 1.0308630362

Subproblem 2

x + 0.2307692308 = -1.261632267 Simplifying x + 0.2307692308 = -1.261632267 Reorder the terms: 0.2307692308 + x = -1.261632267 Solving 0.2307692308 + x = -1.261632267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2307692308' to each side of the equation. 0.2307692308 + -0.2307692308 + x = -1.261632267 + -0.2307692308 Combine like terms: 0.2307692308 + -0.2307692308 = 0.0000000000 0.0000000000 + x = -1.261632267 + -0.2307692308 x = -1.261632267 + -0.2307692308 Combine like terms: -1.261632267 + -0.2307692308 = -1.4924014978 x = -1.4924014978 Simplifying x = -1.4924014978

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0308630362, -1.4924014978}

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