2z^2+8=-9z

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Solution for 2z^2+8=-9z equation:


Simplifying
2z2 + 8 = -9z

Reorder the terms:
8 + 2z2 = -9z

Solving
8 + 2z2 = -9z

Solving for variable 'z'.

Reorder the terms:
8 + 9z + 2z2 = -9z + 9z

Combine like terms: -9z + 9z = 0
8 + 9z + 2z2 = 0

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

Divide each side by '2'.
4 + 4.5z + z2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 4.5z + -4 + z2 = 0 + -4

Reorder the terms:
4 + -4 + 4.5z + z2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 4.5z + z2 = 0 + -4
4.5z + z2 = 0 + -4

Combine like terms: 0 + -4 = -4
4.5z + z2 = -4

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 = -4 + 5.0625

Reorder the terms:
5.0625 + 4.5z + z2 = -4 + 5.0625

Combine like terms: -4 + 5.0625 = 1.0625
5.0625 + 4.5z + z2 = 1.0625

Factor a perfect square on the left side:
(z + 2.25)(z + 2.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(z + 2.25) equal to 1.030776406 and -1.030776406.

Subproblem 1

z + 2.25 = 1.030776406 Simplifying z + 2.25 = 1.030776406 Reorder the terms: 2.25 + z = 1.030776406 Solving 2.25 + z = 1.030776406 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 = 1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = 1.030776406 + -2.25 z = 1.030776406 + -2.25 Combine like terms: 1.030776406 + -2.25 = -1.219223594 z = -1.219223594 Simplifying z = -1.219223594

Subproblem 2

z + 2.25 = -1.030776406 Simplifying z + 2.25 = -1.030776406 Reorder the terms: 2.25 + z = -1.030776406 Solving 2.25 + z = -1.030776406 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 = -1.030776406 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = -1.030776406 + -2.25 z = -1.030776406 + -2.25 Combine like terms: -1.030776406 + -2.25 = -3.280776406 z = -3.280776406 Simplifying z = -3.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-1.219223594, -3.280776406}

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