7z+8=8z^2

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Solution for 7z+8=8z^2 equation:


Simplifying
7z + 8 = 8z2

Reorder the terms:
8 + 7z = 8z2

Solving
8 + 7z = 8z2

Solving for variable 'z'.

Combine like terms: 8z2 + -8z2 = 0
8 + 7z + -8z2 = 0

Begin completing the square.  Divide all terms by
-8 the coefficient of the squared term: 

Divide each side by '-8'.
-1 + -0.875z + z2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -0.875z + 1 + z2 = 0 + 1

Reorder the terms:
-1 + 1 + -0.875z + z2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -0.875z + z2 = 0 + 1
-0.875z + z2 = 0 + 1

Combine like terms: 0 + 1 = 1
-0.875z + z2 = 1

The z term is -0.875z.  Take half its coefficient (-0.4375).
Square it (0.19140625) and add it to both sides.

Add '0.19140625' to each side of the equation.
-0.875z + 0.19140625 + z2 = 1 + 0.19140625

Reorder the terms:
0.19140625 + -0.875z + z2 = 1 + 0.19140625

Combine like terms: 1 + 0.19140625 = 1.19140625
0.19140625 + -0.875z + z2 = 1.19140625

Factor a perfect square on the left side:
(z + -0.4375)(z + -0.4375) = 1.19140625

Calculate the square root of the right side: 1.091515575

Break this problem into two subproblems by setting 
(z + -0.4375) equal to 1.091515575 and -1.091515575.

Subproblem 1

z + -0.4375 = 1.091515575 Simplifying z + -0.4375 = 1.091515575 Reorder the terms: -0.4375 + z = 1.091515575 Solving -0.4375 + z = 1.091515575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + z = 1.091515575 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + z = 1.091515575 + 0.4375 z = 1.091515575 + 0.4375 Combine like terms: 1.091515575 + 0.4375 = 1.529015575 z = 1.529015575 Simplifying z = 1.529015575

Subproblem 2

z + -0.4375 = -1.091515575 Simplifying z + -0.4375 = -1.091515575 Reorder the terms: -0.4375 + z = -1.091515575 Solving -0.4375 + z = -1.091515575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + z = -1.091515575 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + z = -1.091515575 + 0.4375 z = -1.091515575 + 0.4375 Combine like terms: -1.091515575 + 0.4375 = -0.654015575 z = -0.654015575 Simplifying z = -0.654015575

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.529015575, -0.654015575}

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