z^2+10z-22=0

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Solution for z^2+10z-22=0 equation:


Simplifying
z2 + 10z + -22 = 0

Reorder the terms:
-22 + 10z + z2 = 0

Solving
-22 + 10z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + 10z + 22 + z2 = 0 + 22

Reorder the terms:
-22 + 22 + 10z + z2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + 10z + z2 = 0 + 22
10z + z2 = 0 + 22

Combine like terms: 0 + 22 = 22
10z + z2 = 22

The z term is 10z.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10z + 25 + z2 = 22 + 25

Reorder the terms:
25 + 10z + z2 = 22 + 25

Combine like terms: 22 + 25 = 47
25 + 10z + z2 = 47

Factor a perfect square on the left side:
(z + 5)(z + 5) = 47

Calculate the square root of the right side: 6.8556546

Break this problem into two subproblems by setting 
(z + 5) equal to 6.8556546 and -6.8556546.

Subproblem 1

z + 5 = 6.8556546 Simplifying z + 5 = 6.8556546 Reorder the terms: 5 + z = 6.8556546 Solving 5 + z = 6.8556546 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + z = 6.8556546 + -5 Combine like terms: 5 + -5 = 0 0 + z = 6.8556546 + -5 z = 6.8556546 + -5 Combine like terms: 6.8556546 + -5 = 1.8556546 z = 1.8556546 Simplifying z = 1.8556546

Subproblem 2

z + 5 = -6.8556546 Simplifying z + 5 = -6.8556546 Reorder the terms: 5 + z = -6.8556546 Solving 5 + z = -6.8556546 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + z = -6.8556546 + -5 Combine like terms: 5 + -5 = 0 0 + z = -6.8556546 + -5 z = -6.8556546 + -5 Combine like terms: -6.8556546 + -5 = -11.8556546 z = -11.8556546 Simplifying z = -11.8556546

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.8556546, -11.8556546}

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