2z^2+3z-6=0

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Solution for 2z^2+3z-6=0 equation:


Simplifying
2z2 + 3z + -6 = 0

Reorder the terms:
-6 + 3z + 2z2 = 0

Solving
-6 + 3z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
-3 + 1.5z + z2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 1.5z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + 1.5z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 1.5z + z2 = 0 + 3
1.5z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
1.5z + z2 = 3

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 = 3 + 0.5625

Reorder the terms:
0.5625 + 1.5z + z2 = 3 + 0.5625

Combine like terms: 3 + 0.5625 = 3.5625
0.5625 + 1.5z + z2 = 3.5625

Factor a perfect square on the left side:
(z + 0.75)(z + 0.75) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(z + 0.75) equal to 1.887458609 and -1.887458609.

Subproblem 1

z + 0.75 = 1.887458609 Simplifying z + 0.75 = 1.887458609 Reorder the terms: 0.75 + z = 1.887458609 Solving 0.75 + z = 1.887458609 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 = 1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.887458609 + -0.75 z = 1.887458609 + -0.75 Combine like terms: 1.887458609 + -0.75 = 1.137458609 z = 1.137458609 Simplifying z = 1.137458609

Subproblem 2

z + 0.75 = -1.887458609 Simplifying z + 0.75 = -1.887458609 Reorder the terms: 0.75 + z = -1.887458609 Solving 0.75 + z = -1.887458609 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 = -1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.887458609 + -0.75 z = -1.887458609 + -0.75 Combine like terms: -1.887458609 + -0.75 = -2.637458609 z = -2.637458609 Simplifying z = -2.637458609

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.137458609, -2.637458609}

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