4z^2+2z-3=0

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Solution for 4z^2+2z-3=0 equation:


Simplifying
4z2 + 2z + -3 = 0

Reorder the terms:
-3 + 2z + 4z2 = 0

Solving
-3 + 2z + 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'.
-0.75 + 0.5z + z2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 0.5z + 0.75 + z2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 0.5z + z2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 0.5z + z2 = 0 + 0.75
0.5z + z2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
0.5z + z2 = 0.75

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 = 0.75 + 0.0625

Reorder the terms:
0.0625 + 0.5z + z2 = 0.75 + 0.0625

Combine like terms: 0.75 + 0.0625 = 0.8125
0.0625 + 0.5z + z2 = 0.8125

Factor a perfect square on the left side:
(z + 0.25)(z + 0.25) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(z + 0.25) equal to 0.901387819 and -0.901387819.

Subproblem 1

z + 0.25 = 0.901387819 Simplifying z + 0.25 = 0.901387819 Reorder the terms: 0.25 + z = 0.901387819 Solving 0.25 + z = 0.901387819 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 = 0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = 0.901387819 + -0.25 z = 0.901387819 + -0.25 Combine like terms: 0.901387819 + -0.25 = 0.651387819 z = 0.651387819 Simplifying z = 0.651387819

Subproblem 2

z + 0.25 = -0.901387819 Simplifying z + 0.25 = -0.901387819 Reorder the terms: 0.25 + z = -0.901387819 Solving 0.25 + z = -0.901387819 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 = -0.901387819 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = -0.901387819 + -0.25 z = -0.901387819 + -0.25 Combine like terms: -0.901387819 + -0.25 = -1.151387819 z = -1.151387819 Simplifying z = -1.151387819

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.651387819, -1.151387819}

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