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 = 10
25 + -10z + z2 = 10

Factor a perfect square on the left side:
(z + -5)(z + -5) = 10

Calculate the square root of the right side: 3.16227766

Break this problem into two subproblems by setting 
(z + -5) equal to 3.16227766 and -3.16227766.

Subproblem 1

z + -5 = 3.16227766 Simplifying z + -5 = 3.16227766 Reorder the terms: -5 + z = 3.16227766 Solving -5 + z = 3.16227766 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 = 3.16227766 + 5 Combine like terms: -5 + 5 = 0 0 + z = 3.16227766 + 5 z = 3.16227766 + 5 Combine like terms: 3.16227766 + 5 = 8.16227766 z = 8.16227766 Simplifying z = 8.16227766

Subproblem 2

z + -5 = -3.16227766 Simplifying z + -5 = -3.16227766 Reorder the terms: -5 + z = -3.16227766 Solving -5 + z = -3.16227766 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 = -3.16227766 + 5 Combine like terms: -5 + 5 = 0 0 + z = -3.16227766 + 5 z = -3.16227766 + 5 Combine like terms: -3.16227766 + 5 = 1.83772234 z = 1.83772234 Simplifying z = 1.83772234

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.16227766, 1.83772234}

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