4z^2-z=54

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Solution for 4z^2-z=54 equation:


Simplifying
4z2 + -1z = 54

Reorder the terms:
-1z + 4z2 = 54

Solving
-1z + 4z2 = 54

Solving for variable 'z'.

Reorder the terms:
-54 + -1z + 4z2 = 54 + -54

Combine like terms: 54 + -54 = 0
-54 + -1z + 4z2 = 0

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

Divide each side by '4'.
-13.5 + -0.25z + z2 = 0

Move the constant term to the right:

Add '13.5' to each side of the equation.
-13.5 + -0.25z + 13.5 + z2 = 0 + 13.5

Reorder the terms:
-13.5 + 13.5 + -0.25z + z2 = 0 + 13.5

Combine like terms: -13.5 + 13.5 = 0.0
0.0 + -0.25z + z2 = 0 + 13.5
-0.25z + z2 = 0 + 13.5

Combine like terms: 0 + 13.5 = 13.5
-0.25z + z2 = 13.5

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 = 13.5 + 0.015625

Reorder the terms:
0.015625 + -0.25z + z2 = 13.5 + 0.015625

Combine like terms: 13.5 + 0.015625 = 13.515625
0.015625 + -0.25z + z2 = 13.515625

Factor a perfect square on the left side:
(z + -0.125)(z + -0.125) = 13.515625

Calculate the square root of the right side: 3.676360292

Break this problem into two subproblems by setting 
(z + -0.125) equal to 3.676360292 and -3.676360292.

Subproblem 1

z + -0.125 = 3.676360292 Simplifying z + -0.125 = 3.676360292 Reorder the terms: -0.125 + z = 3.676360292 Solving -0.125 + z = 3.676360292 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 = 3.676360292 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = 3.676360292 + 0.125 z = 3.676360292 + 0.125 Combine like terms: 3.676360292 + 0.125 = 3.801360292 z = 3.801360292 Simplifying z = 3.801360292

Subproblem 2

z + -0.125 = -3.676360292 Simplifying z + -0.125 = -3.676360292 Reorder the terms: -0.125 + z = -3.676360292 Solving -0.125 + z = -3.676360292 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 = -3.676360292 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + z = -3.676360292 + 0.125 z = -3.676360292 + 0.125 Combine like terms: -3.676360292 + 0.125 = -3.551360292 z = -3.551360292 Simplifying z = -3.551360292

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.801360292, -3.551360292}

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