z^2+10z=25

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Solution for z^2+10z=25 equation:


Simplifying
z2 + 10z = 25

Reorder the terms:
10z + z2 = 25

Solving
10z + z2 = 25

Solving for variable 'z'.

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

Combine like terms: 25 + -25 = 0
-25 + 10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + 10z + 25 + z2 = 0 + 25

Reorder the terms:
-25 + 25 + 10z + z2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + 10z + z2 = 0 + 25
10z + z2 = 0 + 25

Combine like terms: 0 + 25 = 25
10z + z2 = 25

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 = 25 + 25

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

Combine like terms: 25 + 25 = 50
25 + 10z + z2 = 50

Factor a perfect square on the left side:
(z + 5)(z + 5) = 50

Calculate the square root of the right side: 7.071067812

Break this problem into two subproblems by setting 
(z + 5) equal to 7.071067812 and -7.071067812.

Subproblem 1

z + 5 = 7.071067812 Simplifying z + 5 = 7.071067812 Reorder the terms: 5 + z = 7.071067812 Solving 5 + z = 7.071067812 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 = 7.071067812 + -5 Combine like terms: 5 + -5 = 0 0 + z = 7.071067812 + -5 z = 7.071067812 + -5 Combine like terms: 7.071067812 + -5 = 2.071067812 z = 2.071067812 Simplifying z = 2.071067812

Subproblem 2

z + 5 = -7.071067812 Simplifying z + 5 = -7.071067812 Reorder the terms: 5 + z = -7.071067812 Solving 5 + z = -7.071067812 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 = -7.071067812 + -5 Combine like terms: 5 + -5 = 0 0 + z = -7.071067812 + -5 z = -7.071067812 + -5 Combine like terms: -7.071067812 + -5 = -12.071067812 z = -12.071067812 Simplifying z = -12.071067812

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.071067812, -12.071067812}

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