7z(z-9)-(8z-6)=-16z

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Solution for 7z(z-9)-(8z-6)=-16z equation:


Simplifying
7z(z + -9) + -1(8z + -6) = -16z

Reorder the terms:
7z(-9 + z) + -1(8z + -6) = -16z
(-9 * 7z + z * 7z) + -1(8z + -6) = -16z
(-63z + 7z2) + -1(8z + -6) = -16z

Reorder the terms:
-63z + 7z2 + -1(-6 + 8z) = -16z
-63z + 7z2 + (-6 * -1 + 8z * -1) = -16z
-63z + 7z2 + (6 + -8z) = -16z

Reorder the terms:
6 + -63z + -8z + 7z2 = -16z

Combine like terms: -63z + -8z = -71z
6 + -71z + 7z2 = -16z

Solving
6 + -71z + 7z2 = -16z

Solving for variable 'z'.

Reorder the terms:
6 + -71z + 16z + 7z2 = -16z + 16z

Combine like terms: -71z + 16z = -55z
6 + -55z + 7z2 = -16z + 16z

Combine like terms: -16z + 16z = 0
6 + -55z + 7z2 = 0

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

Divide each side by '7'.
0.8571428571 + -7.857142857z + z2 = 0

Move the constant term to the right:

Add '-0.8571428571' to each side of the equation.
0.8571428571 + -7.857142857z + -0.8571428571 + z2 = 0 + -0.8571428571

Reorder the terms:
0.8571428571 + -0.8571428571 + -7.857142857z + z2 = 0 + -0.8571428571

Combine like terms: 0.8571428571 + -0.8571428571 = 0.0000000000
0.0000000000 + -7.857142857z + z2 = 0 + -0.8571428571
-7.857142857z + z2 = 0 + -0.8571428571

Combine like terms: 0 + -0.8571428571 = -0.8571428571
-7.857142857z + z2 = -0.8571428571

The z term is -7.857142857z.  Take half its coefficient (-3.928571429).
Square it (15.43367347) and add it to both sides.

Add '15.43367347' to each side of the equation.
-7.857142857z + 15.43367347 + z2 = -0.8571428571 + 15.43367347

Reorder the terms:
15.43367347 + -7.857142857z + z2 = -0.8571428571 + 15.43367347

Combine like terms: -0.8571428571 + 15.43367347 = 14.5765306129
15.43367347 + -7.857142857z + z2 = 14.5765306129

Factor a perfect square on the left side:
(z + -3.928571429)(z + -3.928571429) = 14.5765306129

Calculate the square root of the right side: 3.81792229

Break this problem into two subproblems by setting 
(z + -3.928571429) equal to 3.81792229 and -3.81792229.

Subproblem 1

z + -3.928571429 = 3.81792229 Simplifying z + -3.928571429 = 3.81792229 Reorder the terms: -3.928571429 + z = 3.81792229 Solving -3.928571429 + z = 3.81792229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.928571429' to each side of the equation. -3.928571429 + 3.928571429 + z = 3.81792229 + 3.928571429 Combine like terms: -3.928571429 + 3.928571429 = 0.000000000 0.000000000 + z = 3.81792229 + 3.928571429 z = 3.81792229 + 3.928571429 Combine like terms: 3.81792229 + 3.928571429 = 7.746493719 z = 7.746493719 Simplifying z = 7.746493719

Subproblem 2

z + -3.928571429 = -3.81792229 Simplifying z + -3.928571429 = -3.81792229 Reorder the terms: -3.928571429 + z = -3.81792229 Solving -3.928571429 + z = -3.81792229 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.928571429' to each side of the equation. -3.928571429 + 3.928571429 + z = -3.81792229 + 3.928571429 Combine like terms: -3.928571429 + 3.928571429 = 0.000000000 0.000000000 + z = -3.81792229 + 3.928571429 z = -3.81792229 + 3.928571429 Combine like terms: -3.81792229 + 3.928571429 = 0.110649139 z = 0.110649139 Simplifying z = 0.110649139

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.746493719, 0.110649139}

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