4z^2-6z=1

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Solution for 4z^2-6z=1 equation:


Simplifying
4z2 + -6z = 1

Reorder the terms:
-6z + 4z2 = 1

Solving
-6z + 4z2 = 1

Solving for variable 'z'.

Reorder the terms:
-1 + -6z + 4z2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -6z + 4z2 = 0

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

Divide each side by '4'.
-0.25 + -1.5z + z2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1.5z + 0.25 + z2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1.5z + z2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1.5z + z2 = 0 + 0.25
-1.5z + z2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1.5z + z2 = 0.25

The z term is -1.5z.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5z + 0.5625 + z2 = 0.25 + 0.5625

Reorder the terms:
0.5625 + -1.5z + z2 = 0.25 + 0.5625

Combine like terms: 0.25 + 0.5625 = 0.8125
0.5625 + -1.5z + z2 = 0.8125

Factor a perfect square on the left side:
(z + -0.75)(z + -0.75) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(z + -0.75) equal to 0.901387819 and -0.901387819.

Subproblem 1

z + -0.75 = 0.901387819 Simplifying z + -0.75 = 0.901387819 Reorder the terms: -0.75 + z = 0.901387819 Solving -0.75 + z = 0.901387819 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 0.901387819 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 0.901387819 + 0.75 z = 0.901387819 + 0.75 Combine like terms: 0.901387819 + 0.75 = 1.651387819 z = 1.651387819 Simplifying z = 1.651387819

Subproblem 2

z + -0.75 = -0.901387819 Simplifying z + -0.75 = -0.901387819 Reorder the terms: -0.75 + z = -0.901387819 Solving -0.75 + z = -0.901387819 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -0.901387819 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -0.901387819 + 0.75 z = -0.901387819 + 0.75 Combine like terms: -0.901387819 + 0.75 = -0.151387819 z = -0.151387819 Simplifying z = -0.151387819

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.651387819, -0.151387819}

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