13x^2+125x+78=0

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Solution for 13x^2+125x+78=0 equation:


Simplifying
13x2 + 125x + 78 = 0

Reorder the terms:
78 + 125x + 13x2 = 0

Solving
78 + 125x + 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'.
6 + 9.615384615x + x2 = 0

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + 9.615384615x + -6 + x2 = 0 + -6

Reorder the terms:
6 + -6 + 9.615384615x + x2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + 9.615384615x + x2 = 0 + -6
9.615384615x + x2 = 0 + -6

Combine like terms: 0 + -6 = -6
9.615384615x + x2 = -6

The x term is 9.615384615x.  Take half its coefficient (4.807692308).
Square it (23.11390533) and add it to both sides.

Add '23.11390533' to each side of the equation.
9.615384615x + 23.11390533 + x2 = -6 + 23.11390533

Reorder the terms:
23.11390533 + 9.615384615x + x2 = -6 + 23.11390533

Combine like terms: -6 + 23.11390533 = 17.11390533
23.11390533 + 9.615384615x + x2 = 17.11390533

Factor a perfect square on the left side:
(x + 4.807692308)(x + 4.807692308) = 17.11390533

Calculate the square root of the right side: 4.136895615

Break this problem into two subproblems by setting 
(x + 4.807692308) equal to 4.136895615 and -4.136895615.

Subproblem 1

x + 4.807692308 = 4.136895615 Simplifying x + 4.807692308 = 4.136895615 Reorder the terms: 4.807692308 + x = 4.136895615 Solving 4.807692308 + x = 4.136895615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.807692308' to each side of the equation. 4.807692308 + -4.807692308 + x = 4.136895615 + -4.807692308 Combine like terms: 4.807692308 + -4.807692308 = 0.000000000 0.000000000 + x = 4.136895615 + -4.807692308 x = 4.136895615 + -4.807692308 Combine like terms: 4.136895615 + -4.807692308 = -0.670796693 x = -0.670796693 Simplifying x = -0.670796693

Subproblem 2

x + 4.807692308 = -4.136895615 Simplifying x + 4.807692308 = -4.136895615 Reorder the terms: 4.807692308 + x = -4.136895615 Solving 4.807692308 + x = -4.136895615 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.807692308' to each side of the equation. 4.807692308 + -4.807692308 + x = -4.136895615 + -4.807692308 Combine like terms: 4.807692308 + -4.807692308 = 0.000000000 0.000000000 + x = -4.136895615 + -4.807692308 x = -4.136895615 + -4.807692308 Combine like terms: -4.136895615 + -4.807692308 = -8.944587923 x = -8.944587923 Simplifying x = -8.944587923

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.670796693, -8.944587923}

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