z^2-21z-96=0

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Solution for z^2-21z-96=0 equation:


Simplifying
z2 + -21z + -96 = 0

Reorder the terms:
-96 + -21z + z2 = 0

Solving
-96 + -21z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '96' to each side of the equation.
-96 + -21z + 96 + z2 = 0 + 96

Reorder the terms:
-96 + 96 + -21z + z2 = 0 + 96

Combine like terms: -96 + 96 = 0
0 + -21z + z2 = 0 + 96
-21z + z2 = 0 + 96

Combine like terms: 0 + 96 = 96
-21z + z2 = 96

The z term is -21z.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21z + 110.25 + z2 = 96 + 110.25

Reorder the terms:
110.25 + -21z + z2 = 96 + 110.25

Combine like terms: 96 + 110.25 = 206.25
110.25 + -21z + z2 = 206.25

Factor a perfect square on the left side:
(z + -10.5)(z + -10.5) = 206.25

Calculate the square root of the right side: 14.361406616

Break this problem into two subproblems by setting 
(z + -10.5) equal to 14.361406616 and -14.361406616.

Subproblem 1

z + -10.5 = 14.361406616 Simplifying z + -10.5 = 14.361406616 Reorder the terms: -10.5 + z = 14.361406616 Solving -10.5 + z = 14.361406616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + z = 14.361406616 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + z = 14.361406616 + 10.5 z = 14.361406616 + 10.5 Combine like terms: 14.361406616 + 10.5 = 24.861406616 z = 24.861406616 Simplifying z = 24.861406616

Subproblem 2

z + -10.5 = -14.361406616 Simplifying z + -10.5 = -14.361406616 Reorder the terms: -10.5 + z = -14.361406616 Solving -10.5 + z = -14.361406616 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + z = -14.361406616 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + z = -14.361406616 + 10.5 z = -14.361406616 + 10.5 Combine like terms: -14.361406616 + 10.5 = -3.861406616 z = -3.861406616 Simplifying z = -3.861406616

Solution

The solution to the problem is based on the solutions from the subproblems. z = {24.861406616, -3.861406616}

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