z^2-8z=21

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Solution for z^2-8z=21 equation:


Simplifying
z2 + -8z = 21

Reorder the terms:
-8z + z2 = 21

Solving
-8z + z2 = 21

Solving for variable 'z'.

Reorder the terms:
-21 + -8z + z2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + -8z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + -8z + 21 + z2 = 0 + 21

Reorder the terms:
-21 + 21 + -8z + z2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + -8z + z2 = 0 + 21
-8z + z2 = 0 + 21

Combine like terms: 0 + 21 = 21
-8z + z2 = 21

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = 21 + 16

Reorder the terms:
16 + -8z + z2 = 21 + 16

Combine like terms: 21 + 16 = 37
16 + -8z + z2 = 37

Factor a perfect square on the left side:
(z + -4)(z + -4) = 37

Calculate the square root of the right side: 6.08276253

Break this problem into two subproblems by setting 
(z + -4) equal to 6.08276253 and -6.08276253.

Subproblem 1

z + -4 = 6.08276253 Simplifying z + -4 = 6.08276253 Reorder the terms: -4 + z = 6.08276253 Solving -4 + z = 6.08276253 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 6.08276253 + 4 Combine like terms: -4 + 4 = 0 0 + z = 6.08276253 + 4 z = 6.08276253 + 4 Combine like terms: 6.08276253 + 4 = 10.08276253 z = 10.08276253 Simplifying z = 10.08276253

Subproblem 2

z + -4 = -6.08276253 Simplifying z + -4 = -6.08276253 Reorder the terms: -4 + z = -6.08276253 Solving -4 + z = -6.08276253 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -6.08276253 + 4 Combine like terms: -4 + 4 = 0 0 + z = -6.08276253 + 4 z = -6.08276253 + 4 Combine like terms: -6.08276253 + 4 = -2.08276253 z = -2.08276253 Simplifying z = -2.08276253

Solution

The solution to the problem is based on the solutions from the subproblems. z = {10.08276253, -2.08276253}

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