16x^2-31x-5=0

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Solution for 16x^2-31x-5=0 equation:


Simplifying
16x2 + -31x + -5 = 0

Reorder the terms:
-5 + -31x + 16x2 = 0

Solving
-5 + -31x + 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 + -1.9375x + x2 = 0

Move the constant term to the right:

Add '0.3125' to each side of the equation.
-0.3125 + -1.9375x + 0.3125 + x2 = 0 + 0.3125

Reorder the terms:
-0.3125 + 0.3125 + -1.9375x + x2 = 0 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + -1.9375x + x2 = 0 + 0.3125
-1.9375x + x2 = 0 + 0.3125

Combine like terms: 0 + 0.3125 = 0.3125
-1.9375x + x2 = 0.3125

The x term is -1.9375x.  Take half its coefficient (-0.96875).
Square it (0.9384765625) and add it to both sides.

Add '0.9384765625' to each side of the equation.
-1.9375x + 0.9384765625 + x2 = 0.3125 + 0.9384765625

Reorder the terms:
0.9384765625 + -1.9375x + x2 = 0.3125 + 0.9384765625

Combine like terms: 0.3125 + 0.9384765625 = 1.2509765625
0.9384765625 + -1.9375x + x2 = 1.2509765625

Factor a perfect square on the left side:
(x + -0.96875)(x + -0.96875) = 1.2509765625

Calculate the square root of the right side: 1.118470636

Break this problem into two subproblems by setting 
(x + -0.96875) equal to 1.118470636 and -1.118470636.

Subproblem 1

x + -0.96875 = 1.118470636 Simplifying x + -0.96875 = 1.118470636 Reorder the terms: -0.96875 + x = 1.118470636 Solving -0.96875 + x = 1.118470636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = 1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = 1.118470636 + 0.96875 x = 1.118470636 + 0.96875 Combine like terms: 1.118470636 + 0.96875 = 2.087220636 x = 2.087220636 Simplifying x = 2.087220636

Subproblem 2

x + -0.96875 = -1.118470636 Simplifying x + -0.96875 = -1.118470636 Reorder the terms: -0.96875 + x = -1.118470636 Solving -0.96875 + x = -1.118470636 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.96875' to each side of the equation. -0.96875 + 0.96875 + x = -1.118470636 + 0.96875 Combine like terms: -0.96875 + 0.96875 = 0.00000 0.00000 + x = -1.118470636 + 0.96875 x = -1.118470636 + 0.96875 Combine like terms: -1.118470636 + 0.96875 = -0.149720636 x = -0.149720636 Simplifying x = -0.149720636

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.087220636, -0.149720636}

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