z^2+5z-2=0

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


Simplifying
z2 + 5z + -2 = 0

Reorder the terms:
-2 + 5z + z2 = 0

Solving
-2 + 5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 5z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + 5z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 5z + z2 = 0 + 2
5z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
5z + z2 = 2

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 = 2 + 6.25

Reorder the terms:
6.25 + 5z + z2 = 2 + 6.25

Combine like terms: 2 + 6.25 = 8.25
6.25 + 5z + z2 = 8.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(z + 2.5) equal to 2.872281323 and -2.872281323.

Subproblem 1

z + 2.5 = 2.872281323 Simplifying z + 2.5 = 2.872281323 Reorder the terms: 2.5 + z = 2.872281323 Solving 2.5 + z = 2.872281323 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 = 2.872281323 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 2.872281323 + -2.5 z = 2.872281323 + -2.5 Combine like terms: 2.872281323 + -2.5 = 0.372281323 z = 0.372281323 Simplifying z = 0.372281323

Subproblem 2

z + 2.5 = -2.872281323 Simplifying z + 2.5 = -2.872281323 Reorder the terms: 2.5 + z = -2.872281323 Solving 2.5 + z = -2.872281323 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 = -2.872281323 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -2.872281323 + -2.5 z = -2.872281323 + -2.5 Combine like terms: -2.872281323 + -2.5 = -5.372281323 z = -5.372281323 Simplifying z = -5.372281323

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.372281323, -5.372281323}

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