10z(1-z)=0.9

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Solution for 10z(1-z)=0.9 equation:


Simplifying
10z(1 + -1z) = 0.9
(1 * 10z + -1z * 10z) = 0.9
(10z + -10z2) = 0.9

Solving
10z + -10z2 = 0.9

Solving for variable 'z'.

Reorder the terms:
-0.9 + 10z + -10z2 = 0.9 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
-0.9 + 10z + -10z2 = 0.0

Begin completing the square.  Divide all terms by
-10 the coefficient of the squared term: 

Divide each side by '-10'.
0.09 + -1z + z2 = 0.00

Move the constant term to the right:

Add '-0.09' to each side of the equation.
0.09 + -1z + -0.09 + z2 = 0.00 + -0.09

Reorder the terms:
0.09 + -0.09 + -1z + z2 = 0.00 + -0.09

Combine like terms: 0.09 + -0.09 = 0.00
0.00 + -1z + z2 = 0.00 + -0.09
-1z + z2 = 0.00 + -0.09

Combine like terms: 0.00 + -0.09 = -0.09
-1z + z2 = -0.09

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = -0.09 + 0.25

Reorder the terms:
0.25 + -1z + z2 = -0.09 + 0.25

Combine like terms: -0.09 + 0.25 = 0.16
0.25 + -1z + z2 = 0.16

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 0.16

Calculate the square root of the right side: 0.4

Break this problem into two subproblems by setting 
(z + -0.5) equal to 0.4 and -0.4.

Subproblem 1

z + -0.5 = 0.4 Simplifying z + -0.5 = 0.4 Reorder the terms: -0.5 + z = 0.4 Solving -0.5 + z = 0.4 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = 0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.4 + 0.5 z = 0.4 + 0.5 Combine like terms: 0.4 + 0.5 = 0.9 z = 0.9 Simplifying z = 0.9

Subproblem 2

z + -0.5 = -0.4 Simplifying z + -0.5 = -0.4 Reorder the terms: -0.5 + z = -0.4 Solving -0.5 + z = -0.4 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + z = -0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.4 + 0.5 z = -0.4 + 0.5 Combine like terms: -0.4 + 0.5 = 0.1 z = 0.1 Simplifying z = 0.1

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.9, 0.1}

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