z^2-5z+3=0

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Solution for z^2-5z+3=0 equation:


Simplifying
z2 + -5z + 3 = 0

Reorder the terms:
3 + -5z + z2 = 0

Solving
3 + -5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -5z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + -5z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -5z + z2 = 0 + -3
-5z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
-5z + z2 = -3

The z term is -5z.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5z + 6.25 + z2 = -3 + 6.25

Reorder the terms:
6.25 + -5z + z2 = -3 + 6.25

Combine like terms: -3 + 6.25 = 3.25
6.25 + -5z + z2 = 3.25

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + -2.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

z + -2.5 = 1.802775638 Simplifying z + -2.5 = 1.802775638 Reorder the terms: -2.5 + z = 1.802775638 Solving -2.5 + z = 1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 1.802775638 + 2.5 z = 1.802775638 + 2.5 Combine like terms: 1.802775638 + 2.5 = 4.302775638 z = 4.302775638 Simplifying z = 4.302775638

Subproblem 2

z + -2.5 = -1.802775638 Simplifying z + -2.5 = -1.802775638 Reorder the terms: -2.5 + z = -1.802775638 Solving -2.5 + z = -1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -1.802775638 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -1.802775638 + 2.5 z = -1.802775638 + 2.5 Combine like terms: -1.802775638 + 2.5 = 0.697224362 z = 0.697224362 Simplifying z = 0.697224362

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.302775638, 0.697224362}

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