z^2+10z=15

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Solution for z^2+10z=15 equation:


Simplifying
z2 + 10z = 15

Reorder the terms:
10z + z2 = 15

Solving
10z + z2 = 15

Solving for variable 'z'.

Reorder the terms:
-15 + 10z + z2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 10z + 15 + z2 = 0 + 15

Reorder the terms:
-15 + 15 + 10z + z2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 10z + z2 = 0 + 15
10z + z2 = 0 + 15

Combine like terms: 0 + 15 = 15
10z + z2 = 15

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 = 15 + 25

Reorder the terms:
25 + 10z + z2 = 15 + 25

Combine like terms: 15 + 25 = 40
25 + 10z + z2 = 40

Factor a perfect square on the left side:
(z + 5)(z + 5) = 40

Calculate the square root of the right side: 6.32455532

Break this problem into two subproblems by setting 
(z + 5) equal to 6.32455532 and -6.32455532.

Subproblem 1

z + 5 = 6.32455532 Simplifying z + 5 = 6.32455532 Reorder the terms: 5 + z = 6.32455532 Solving 5 + z = 6.32455532 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 = 6.32455532 + -5 Combine like terms: 5 + -5 = 0 0 + z = 6.32455532 + -5 z = 6.32455532 + -5 Combine like terms: 6.32455532 + -5 = 1.32455532 z = 1.32455532 Simplifying z = 1.32455532

Subproblem 2

z + 5 = -6.32455532 Simplifying z + 5 = -6.32455532 Reorder the terms: 5 + z = -6.32455532 Solving 5 + z = -6.32455532 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 = -6.32455532 + -5 Combine like terms: 5 + -5 = 0 0 + z = -6.32455532 + -5 z = -6.32455532 + -5 Combine like terms: -6.32455532 + -5 = -11.32455532 z = -11.32455532 Simplifying z = -11.32455532

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.32455532, -11.32455532}

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