16x^2+13x-5=0

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Solution for 16x^2+13x-5=0 equation:


Simplifying
16x2 + 13x + -5 = 0

Reorder the terms:
-5 + 13x + 16x2 = 0

Solving
-5 + 13x + 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'.
-0.3125 + 0.8125x + x2 = 0

Move the constant term to the right:

Add '0.3125' to each side of the equation.
-0.3125 + 0.8125x + 0.3125 + x2 = 0 + 0.3125

Reorder the terms:
-0.3125 + 0.3125 + 0.8125x + x2 = 0 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + 0.8125x + x2 = 0 + 0.3125
0.8125x + x2 = 0 + 0.3125

Combine like terms: 0 + 0.3125 = 0.3125
0.8125x + x2 = 0.3125

The x term is 0.8125x.  Take half its coefficient (0.40625).
Square it (0.1650390625) and add it to both sides.

Add '0.1650390625' to each side of the equation.
0.8125x + 0.1650390625 + x2 = 0.3125 + 0.1650390625

Reorder the terms:
0.1650390625 + 0.8125x + x2 = 0.3125 + 0.1650390625

Combine like terms: 0.3125 + 0.1650390625 = 0.4775390625
0.1650390625 + 0.8125x + x2 = 0.4775390625

Factor a perfect square on the left side:
(x + 0.40625)(x + 0.40625) = 0.4775390625

Calculate the square root of the right side: 0.691042012

Break this problem into two subproblems by setting 
(x + 0.40625) equal to 0.691042012 and -0.691042012.

Subproblem 1

x + 0.40625 = 0.691042012 Simplifying x + 0.40625 = 0.691042012 Reorder the terms: 0.40625 + x = 0.691042012 Solving 0.40625 + x = 0.691042012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.40625' to each side of the equation. 0.40625 + -0.40625 + x = 0.691042012 + -0.40625 Combine like terms: 0.40625 + -0.40625 = 0.00000 0.00000 + x = 0.691042012 + -0.40625 x = 0.691042012 + -0.40625 Combine like terms: 0.691042012 + -0.40625 = 0.284792012 x = 0.284792012 Simplifying x = 0.284792012

Subproblem 2

x + 0.40625 = -0.691042012 Simplifying x + 0.40625 = -0.691042012 Reorder the terms: 0.40625 + x = -0.691042012 Solving 0.40625 + x = -0.691042012 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.40625' to each side of the equation. 0.40625 + -0.40625 + x = -0.691042012 + -0.40625 Combine like terms: 0.40625 + -0.40625 = 0.00000 0.00000 + x = -0.691042012 + -0.40625 x = -0.691042012 + -0.40625 Combine like terms: -0.691042012 + -0.40625 = -1.097292012 x = -1.097292012 Simplifying x = -1.097292012

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.284792012, -1.097292012}

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