16z^2+8z-5=0

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Solution for 16z^2+8z-5=0 equation:


Simplifying
16z2 + 8z + -5 = 0

Reorder the terms:
-5 + 8z + 16z2 = 0

Solving
-5 + 8z + 16z2 = 0

Solving for variable 'z'.

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

Divide each side by '16'.
-0.3125 + 0.5z + z2 = 0

Move the constant term to the right:

Add '0.3125' to each side of the equation.
-0.3125 + 0.5z + 0.3125 + z2 = 0 + 0.3125

Reorder the terms:
-0.3125 + 0.3125 + 0.5z + z2 = 0 + 0.3125

Combine like terms: -0.3125 + 0.3125 = 0.0000
0.0000 + 0.5z + z2 = 0 + 0.3125
0.5z + z2 = 0 + 0.3125

Combine like terms: 0 + 0.3125 = 0.3125
0.5z + z2 = 0.3125

The z term is 0.5z.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5z + 0.0625 + z2 = 0.3125 + 0.0625

Reorder the terms:
0.0625 + 0.5z + z2 = 0.3125 + 0.0625

Combine like terms: 0.3125 + 0.0625 = 0.375
0.0625 + 0.5z + z2 = 0.375

Factor a perfect square on the left side:
(z + 0.25)(z + 0.25) = 0.375

Calculate the square root of the right side: 0.612372436

Break this problem into two subproblems by setting 
(z + 0.25) equal to 0.612372436 and -0.612372436.

Subproblem 1

z + 0.25 = 0.612372436 Simplifying z + 0.25 = 0.612372436 Reorder the terms: 0.25 + z = 0.612372436 Solving 0.25 + z = 0.612372436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = 0.612372436 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = 0.612372436 + -0.25 z = 0.612372436 + -0.25 Combine like terms: 0.612372436 + -0.25 = 0.362372436 z = 0.362372436 Simplifying z = 0.362372436

Subproblem 2

z + 0.25 = -0.612372436 Simplifying z + 0.25 = -0.612372436 Reorder the terms: 0.25 + z = -0.612372436 Solving 0.25 + z = -0.612372436 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = -0.612372436 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = -0.612372436 + -0.25 z = -0.612372436 + -0.25 Combine like terms: -0.612372436 + -0.25 = -0.862372436 z = -0.862372436 Simplifying z = -0.862372436

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.362372436, -0.862372436}

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