z^2+3z-6=0

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


Simplifying
z2 + 3z + -6 = 0

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

Solving
-6 + 3z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 3z + 6 + z2 = 0 + 6

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

Combine like terms: -6 + 6 = 0
0 + 3z + z2 = 0 + 6
3z + z2 = 0 + 6

Combine like terms: 0 + 6 = 6
3z + z2 = 6

The z term is 3z.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3z + 2.25 + z2 = 6 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 6 + 2.25

Combine like terms: 6 + 2.25 = 8.25
2.25 + 3z + z2 = 8.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(z + 1.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

z + 1.5 = 2.872281323 Simplifying z + 1.5 = 2.872281323 Reorder the terms: 1.5 + z = 2.872281323 Solving 1.5 + z = 2.872281323 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = 2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 2.872281323 + -1.5 z = 2.872281323 + -1.5 Combine like terms: 2.872281323 + -1.5 = 1.372281323 z = 1.372281323 Simplifying z = 1.372281323

Subproblem 2

z + 1.5 = -2.872281323 Simplifying z + 1.5 = -2.872281323 Reorder the terms: 1.5 + z = -2.872281323 Solving 1.5 + z = -2.872281323 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + z = -2.872281323 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -2.872281323 + -1.5 z = -2.872281323 + -1.5 Combine like terms: -2.872281323 + -1.5 = -4.372281323 z = -4.372281323 Simplifying z = -4.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.372281323, -4.372281323}

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