4z^2+3z-2=0

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


Simplifying
4z2 + 3z + -2 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.75z + z2 = 0 + 0.5
0.75z + z2 = 0 + 0.5

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

The z term is 0.75z.  Take half its coefficient (0.375).
Square it (0.140625) and add it to both sides.

Add '0.140625' to each side of the equation.
0.75z + 0.140625 + z2 = 0.5 + 0.140625

Reorder the terms:
0.140625 + 0.75z + z2 = 0.5 + 0.140625

Combine like terms: 0.5 + 0.140625 = 0.640625
0.140625 + 0.75z + z2 = 0.640625

Factor a perfect square on the left side:
(z + 0.375)(z + 0.375) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(z + 0.375) equal to 0.80039053 and -0.80039053.

Subproblem 1

z + 0.375 = 0.80039053 Simplifying z + 0.375 = 0.80039053 Reorder the terms: 0.375 + z = 0.80039053 Solving 0.375 + z = 0.80039053 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = 0.80039053 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = 0.80039053 + -0.375 z = 0.80039053 + -0.375 Combine like terms: 0.80039053 + -0.375 = 0.42539053 z = 0.42539053 Simplifying z = 0.42539053

Subproblem 2

z + 0.375 = -0.80039053 Simplifying z + 0.375 = -0.80039053 Reorder the terms: 0.375 + z = -0.80039053 Solving 0.375 + z = -0.80039053 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = -0.80039053 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = -0.80039053 + -0.375 z = -0.80039053 + -0.375 Combine like terms: -0.80039053 + -0.375 = -1.17539053 z = -1.17539053 Simplifying z = -1.17539053

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.42539053, -1.17539053}

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