z^2+3z-5=0

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


Simplifying
z2 + 3z + -5 = 0

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

Solving
-5 + 3z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

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

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

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

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

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 = 5 + 2.25

Reorder the terms:
2.25 + 3z + z2 = 5 + 2.25

Combine like terms: 5 + 2.25 = 7.25
2.25 + 3z + z2 = 7.25

Factor a perfect square on the left side:
(z + 1.5)(z + 1.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(z + 1.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

z + 1.5 = 2.692582404 Simplifying z + 1.5 = 2.692582404 Reorder the terms: 1.5 + z = 2.692582404 Solving 1.5 + z = 2.692582404 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 = 2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = 2.692582404 + -1.5 z = 2.692582404 + -1.5 Combine like terms: 2.692582404 + -1.5 = 1.192582404 z = 1.192582404 Simplifying z = 1.192582404

Subproblem 2

z + 1.5 = -2.692582404 Simplifying z + 1.5 = -2.692582404 Reorder the terms: 1.5 + z = -2.692582404 Solving 1.5 + z = -2.692582404 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 = -2.692582404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + z = -2.692582404 + -1.5 z = -2.692582404 + -1.5 Combine like terms: -2.692582404 + -1.5 = -4.192582404 z = -4.192582404 Simplifying z = -4.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.192582404, -4.192582404}

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