16x^2+41x+14=0

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Solution for 16x^2+41x+14=0 equation:


Simplifying
16x2 + 41x + 14 = 0

Reorder the terms:
14 + 41x + 16x2 = 0

Solving
14 + 41x + 16x2 = 0

Solving for variable 'x'.

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

Divide each side by '16'.
0.875 + 2.5625x + x2 = 0

Move the constant term to the right:

Add '-0.875' to each side of the equation.
0.875 + 2.5625x + -0.875 + x2 = 0 + -0.875

Reorder the terms:
0.875 + -0.875 + 2.5625x + x2 = 0 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + 2.5625x + x2 = 0 + -0.875
2.5625x + x2 = 0 + -0.875

Combine like terms: 0 + -0.875 = -0.875
2.5625x + x2 = -0.875

The x term is 2.5625x.  Take half its coefficient (1.28125).
Square it (1.641601563) and add it to both sides.

Add '1.641601563' to each side of the equation.
2.5625x + 1.641601563 + x2 = -0.875 + 1.641601563

Reorder the terms:
1.641601563 + 2.5625x + x2 = -0.875 + 1.641601563

Combine like terms: -0.875 + 1.641601563 = 0.766601563
1.641601563 + 2.5625x + x2 = 0.766601563

Factor a perfect square on the left side:
(x + 1.28125)(x + 1.28125) = 0.766601563

Calculate the square root of the right side: 0.875557858

Break this problem into two subproblems by setting 
(x + 1.28125) equal to 0.875557858 and -0.875557858.

Subproblem 1

x + 1.28125 = 0.875557858 Simplifying x + 1.28125 = 0.875557858 Reorder the terms: 1.28125 + x = 0.875557858 Solving 1.28125 + x = 0.875557858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.28125' to each side of the equation. 1.28125 + -1.28125 + x = 0.875557858 + -1.28125 Combine like terms: 1.28125 + -1.28125 = 0.00000 0.00000 + x = 0.875557858 + -1.28125 x = 0.875557858 + -1.28125 Combine like terms: 0.875557858 + -1.28125 = -0.405692142 x = -0.405692142 Simplifying x = -0.405692142

Subproblem 2

x + 1.28125 = -0.875557858 Simplifying x + 1.28125 = -0.875557858 Reorder the terms: 1.28125 + x = -0.875557858 Solving 1.28125 + x = -0.875557858 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.28125' to each side of the equation. 1.28125 + -1.28125 + x = -0.875557858 + -1.28125 Combine like terms: 1.28125 + -1.28125 = 0.00000 0.00000 + x = -0.875557858 + -1.28125 x = -0.875557858 + -1.28125 Combine like terms: -0.875557858 + -1.28125 = -2.156807858 x = -2.156807858 Simplifying x = -2.156807858

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-0.405692142, -2.156807858}

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