z^2=-3z+3

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


Simplifying
z2 = -3z + 3

Reorder the terms:
z2 = 3 + -3z

Solving
z2 = 3 + -3z

Solving for variable 'z'.

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

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

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

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

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 3 + 2.25

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

Combine like terms: 3 + 2.25 = 5.25
2.25 + 3z + z2 = 5.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(z + 1.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

z + 1.5 = 2.291287847 Simplifying z + 1.5 = 2.291287847 Reorder the terms: 1.5 + z = 2.291287847 Solving 1.5 + z = 2.291287847 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 = 2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 2.291287847 + -1.5 z = 2.291287847 + -1.5 Combine like terms: 2.291287847 + -1.5 = 0.791287847 z = 0.791287847 Simplifying z = 0.791287847

Subproblem 2

z + 1.5 = -2.291287847 Simplifying z + 1.5 = -2.291287847 Reorder the terms: 1.5 + z = -2.291287847 Solving 1.5 + z = -2.291287847 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 = -2.291287847 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -2.291287847 + -1.5 z = -2.291287847 + -1.5 Combine like terms: -2.291287847 + -1.5 = -3.791287847 z = -3.791287847 Simplifying z = -3.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.791287847, -3.791287847}

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