z^2-10z=-13

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Solution for z^2-10z=-13 equation:


Simplifying
z2 + -10z = -13

Reorder the terms:
-10z + z2 = -13

Solving
-10z + z2 = -13

Solving for variable 'z'.

Reorder the terms:
13 + -10z + z2 = -13 + 13

Combine like terms: -13 + 13 = 0
13 + -10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-13' to each side of the equation.
13 + -10z + -13 + z2 = 0 + -13

Reorder the terms:
13 + -13 + -10z + z2 = 0 + -13

Combine like terms: 13 + -13 = 0
0 + -10z + z2 = 0 + -13
-10z + z2 = 0 + -13

Combine like terms: 0 + -13 = -13
-10z + z2 = -13

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 = -13 + 25

Reorder the terms:
25 + -10z + z2 = -13 + 25

Combine like terms: -13 + 25 = 12
25 + -10z + z2 = 12

Factor a perfect square on the left side:
(z + -5)(z + -5) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(z + -5) equal to 3.464101615 and -3.464101615.

Subproblem 1

z + -5 = 3.464101615 Simplifying z + -5 = 3.464101615 Reorder the terms: -5 + z = 3.464101615 Solving -5 + z = 3.464101615 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 = 3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + z = 3.464101615 + 5 z = 3.464101615 + 5 Combine like terms: 3.464101615 + 5 = 8.464101615 z = 8.464101615 Simplifying z = 8.464101615

Subproblem 2

z + -5 = -3.464101615 Simplifying z + -5 = -3.464101615 Reorder the terms: -5 + z = -3.464101615 Solving -5 + z = -3.464101615 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 = -3.464101615 + 5 Combine like terms: -5 + 5 = 0 0 + z = -3.464101615 + 5 z = -3.464101615 + 5 Combine like terms: -3.464101615 + 5 = 1.535898385 z = 1.535898385 Simplifying z = 1.535898385

Solution

The solution to the problem is based on the solutions from the subproblems. z = {8.464101615, 1.535898385}

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