z^2-16z=-9

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Solution for z^2-16z=-9 equation:


Simplifying
z2 + -16z = -9

Reorder the terms:
-16z + z2 = -9

Solving
-16z + z2 = -9

Solving for variable 'z'.

Reorder the terms:
9 + -16z + z2 = -9 + 9

Combine like terms: -9 + 9 = 0
9 + -16z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -16z + -9 + z2 = 0 + -9

Reorder the terms:
9 + -9 + -16z + z2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -16z + z2 = 0 + -9
-16z + z2 = 0 + -9

Combine like terms: 0 + -9 = -9
-16z + z2 = -9

The z term is -16z.  Take half its coefficient (-8).
Square it (64) and add it to both sides.

Add '64' to each side of the equation.
-16z + 64 + z2 = -9 + 64

Reorder the terms:
64 + -16z + z2 = -9 + 64

Combine like terms: -9 + 64 = 55
64 + -16z + z2 = 55

Factor a perfect square on the left side:
(z + -8)(z + -8) = 55

Calculate the square root of the right side: 7.416198487

Break this problem into two subproblems by setting 
(z + -8) equal to 7.416198487 and -7.416198487.

Subproblem 1

z + -8 = 7.416198487 Simplifying z + -8 = 7.416198487 Reorder the terms: -8 + z = 7.416198487 Solving -8 + z = 7.416198487 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = 7.416198487 + 8 Combine like terms: -8 + 8 = 0 0 + z = 7.416198487 + 8 z = 7.416198487 + 8 Combine like terms: 7.416198487 + 8 = 15.416198487 z = 15.416198487 Simplifying z = 15.416198487

Subproblem 2

z + -8 = -7.416198487 Simplifying z + -8 = -7.416198487 Reorder the terms: -8 + z = -7.416198487 Solving -8 + z = -7.416198487 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + z = -7.416198487 + 8 Combine like terms: -8 + 8 = 0 0 + z = -7.416198487 + 8 z = -7.416198487 + 8 Combine like terms: -7.416198487 + 8 = 0.583801513 z = 0.583801513 Simplifying z = 0.583801513

Solution

The solution to the problem is based on the solutions from the subproblems. z = {15.416198487, 0.583801513}

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