3z^2+2z-3=0

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


Simplifying
3z2 + 2z + -3 = 0

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

Solving
-3 + 2z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
-1 + 0.6666666667z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.6666666667z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.6666666667z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.6666666667z + z2 = 0 + 1
0.6666666667z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.6666666667z + z2 = 1

The z term is 0.6666666667z.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667z + 0.1111111112 + z2 = 1 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667z + z2 = 1 + 0.1111111112

Combine like terms: 1 + 0.1111111112 = 1.1111111112
0.1111111112 + 0.6666666667z + z2 = 1.1111111112

Factor a perfect square on the left side:
(z + 0.3333333334)(z + 0.3333333334) = 1.1111111112

Calculate the square root of the right side: 1.054092553

Break this problem into two subproblems by setting 
(z + 0.3333333334) equal to 1.054092553 and -1.054092553.

Subproblem 1

z + 0.3333333334 = 1.054092553 Simplifying z + 0.3333333334 = 1.054092553 Reorder the terms: 0.3333333334 + z = 1.054092553 Solving 0.3333333334 + z = 1.054092553 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = 1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = 1.054092553 + -0.3333333334 z = 1.054092553 + -0.3333333334 Combine like terms: 1.054092553 + -0.3333333334 = 0.7207592196 z = 0.7207592196 Simplifying z = 0.7207592196

Subproblem 2

z + 0.3333333334 = -1.054092553 Simplifying z + 0.3333333334 = -1.054092553 Reorder the terms: 0.3333333334 + z = -1.054092553 Solving 0.3333333334 + z = -1.054092553 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = -1.054092553 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = -1.054092553 + -0.3333333334 z = -1.054092553 + -0.3333333334 Combine like terms: -1.054092553 + -0.3333333334 = -1.3874258864 z = -1.3874258864 Simplifying z = -1.3874258864

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.7207592196, -1.3874258864}

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