2z^2+9z-4=0

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Solution for 2z^2+9z-4=0 equation:


Simplifying
2z2 + 9z + -4 = 0

Reorder the terms:
-4 + 9z + 2z2 = 0

Solving
-4 + 9z + 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'.
-2 + 4.5z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 4.5z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + 4.5z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 4.5z + z2 = 0 + 2
4.5z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
4.5z + z2 = 2

The z term is 4.5z.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5z + 5.0625 + z2 = 2 + 5.0625

Reorder the terms:
5.0625 + 4.5z + z2 = 2 + 5.0625

Combine like terms: 2 + 5.0625 = 7.0625
5.0625 + 4.5z + z2 = 7.0625

Factor a perfect square on the left side:
(z + 2.25)(z + 2.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(z + 2.25) equal to 2.657536453 and -2.657536453.

Subproblem 1

z + 2.25 = 2.657536453 Simplifying z + 2.25 = 2.657536453 Reorder the terms: 2.25 + z = 2.657536453 Solving 2.25 + z = 2.657536453 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = 2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = 2.657536453 + -2.25 z = 2.657536453 + -2.25 Combine like terms: 2.657536453 + -2.25 = 0.407536453 z = 0.407536453 Simplifying z = 0.407536453

Subproblem 2

z + 2.25 = -2.657536453 Simplifying z + 2.25 = -2.657536453 Reorder the terms: 2.25 + z = -2.657536453 Solving 2.25 + z = -2.657536453 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = -2.657536453 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = -2.657536453 + -2.25 z = -2.657536453 + -2.25 Combine like terms: -2.657536453 + -2.25 = -4.907536453 z = -4.907536453 Simplifying z = -4.907536453

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.407536453, -4.907536453}

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