15z-z^2-20=0

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Solution for 15z-z^2-20=0 equation:


Simplifying
15z + -1z2 + -20 = 0

Reorder the terms:
-20 + 15z + -1z2 = 0

Solving
-20 + 15z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
20 + -15z + z2 = 0

Move the constant term to the right:

Add '-20' to each side of the equation.
20 + -15z + -20 + z2 = 0 + -20

Reorder the terms:
20 + -20 + -15z + z2 = 0 + -20

Combine like terms: 20 + -20 = 0
0 + -15z + z2 = 0 + -20
-15z + z2 = 0 + -20

Combine like terms: 0 + -20 = -20
-15z + z2 = -20

The z term is -15z.  Take half its coefficient (-7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
-15z + 56.25 + z2 = -20 + 56.25

Reorder the terms:
56.25 + -15z + z2 = -20 + 56.25

Combine like terms: -20 + 56.25 = 36.25
56.25 + -15z + z2 = 36.25

Factor a perfect square on the left side:
(z + -7.5)(z + -7.5) = 36.25

Calculate the square root of the right side: 6.020797289

Break this problem into two subproblems by setting 
(z + -7.5) equal to 6.020797289 and -6.020797289.

Subproblem 1

z + -7.5 = 6.020797289 Simplifying z + -7.5 = 6.020797289 Reorder the terms: -7.5 + z = 6.020797289 Solving -7.5 + z = 6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = 6.020797289 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = 6.020797289 + 7.5 z = 6.020797289 + 7.5 Combine like terms: 6.020797289 + 7.5 = 13.520797289 z = 13.520797289 Simplifying z = 13.520797289

Subproblem 2

z + -7.5 = -6.020797289 Simplifying z + -7.5 = -6.020797289 Reorder the terms: -7.5 + z = -6.020797289 Solving -7.5 + z = -6.020797289 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + z = -6.020797289 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + z = -6.020797289 + 7.5 z = -6.020797289 + 7.5 Combine like terms: -6.020797289 + 7.5 = 1.479202711 z = 1.479202711 Simplifying z = 1.479202711

Solution

The solution to the problem is based on the solutions from the subproblems. z = {13.520797289, 1.479202711}

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