4z^2-z-0.5=0

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Solution for 4z^2-z-0.5=0 equation:


Simplifying
4z2 + -1z + -0.5 = 0

Reorder the terms:
-0.5 + -1z + 4z2 = 0

Solving
-0.5 + -1z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
-0.125 + -0.25z + z2 = 0

Move the constant term to the right:

Add '0.125' to each side of the equation.
-0.125 + -0.25z + 0.125 + z2 = 0 + 0.125

Reorder the terms:
-0.125 + 0.125 + -0.25z + z2 = 0 + 0.125

Combine like terms: -0.125 + 0.125 = 0.000
0.000 + -0.25z + z2 = 0 + 0.125
-0.25z + z2 = 0 + 0.125

Combine like terms: 0 + 0.125 = 0.125
-0.25z + z2 = 0.125

The z term is -0.25z.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25z + 0.015625 + z2 = 0.125 + 0.015625

Reorder the terms:
0.015625 + -0.25z + z2 = 0.125 + 0.015625

Combine like terms: 0.125 + 0.015625 = 0.140625
0.015625 + -0.25z + z2 = 0.140625

Factor a perfect square on the left side:
(z + -0.125)(z + -0.125) = 0.140625

Calculate the square root of the right side: 0.375

Break this problem into two subproblems by setting 
(z + -0.125) equal to 0.375 and -0.375.

Subproblem 1

z + -0.125 = 0.375 Simplifying z + -0.125 = 0.375 Reorder the terms: -0.125 + z = 0.375 Solving -0.125 + z = 0.375 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + z = 0.375 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = 0.375 + 0.125 z = 0.375 + 0.125 Combine like terms: 0.375 + 0.125 = 0.5 z = 0.5 Simplifying z = 0.5

Subproblem 2

z + -0.125 = -0.375 Simplifying z + -0.125 = -0.375 Reorder the terms: -0.125 + z = -0.375 Solving -0.125 + z = -0.375 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + z = -0.375 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = -0.375 + 0.125 z = -0.375 + 0.125 Combine like terms: -0.375 + 0.125 = -0.25 z = -0.25 Simplifying z = -0.25

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.5, -0.25}

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