z^2-10z=-19

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Solution for z^2-10z=-19 equation:


Simplifying
z2 + -10z = -19

Reorder the terms:
-10z + z2 = -19

Solving
-10z + z2 = -19

Solving for variable 'z'.

Reorder the terms:
19 + -10z + z2 = -19 + 19

Combine like terms: -19 + 19 = 0
19 + -10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -10z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + -10z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -10z + z2 = 0 + -19
-10z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
-10z + z2 = -19

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 = -19 + 25

Reorder the terms:
25 + -10z + z2 = -19 + 25

Combine like terms: -19 + 25 = 6
25 + -10z + z2 = 6

Factor a perfect square on the left side:
(z + -5)(z + -5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(z + -5) equal to 2.449489743 and -2.449489743.

Subproblem 1

z + -5 = 2.449489743 Simplifying z + -5 = 2.449489743 Reorder the terms: -5 + z = 2.449489743 Solving -5 + z = 2.449489743 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 = 2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + z = 2.449489743 + 5 z = 2.449489743 + 5 Combine like terms: 2.449489743 + 5 = 7.449489743 z = 7.449489743 Simplifying z = 7.449489743

Subproblem 2

z + -5 = -2.449489743 Simplifying z + -5 = -2.449489743 Reorder the terms: -5 + z = -2.449489743 Solving -5 + z = -2.449489743 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 = -2.449489743 + 5 Combine like terms: -5 + 5 = 0 0 + z = -2.449489743 + 5 z = -2.449489743 + 5 Combine like terms: -2.449489743 + 5 = 2.550510257 z = 2.550510257 Simplifying z = 2.550510257

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.449489743, 2.550510257}

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