5z^2-7=8z

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Solution for 5z^2-7=8z equation:


Simplifying
5z2 + -7 = 8z

Reorder the terms:
-7 + 5z2 = 8z

Solving
-7 + 5z2 = 8z

Solving for variable 'z'.

Reorder the terms:
-7 + -8z + 5z2 = 8z + -8z

Combine like terms: 8z + -8z = 0
-7 + -8z + 5z2 = 0

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

Divide each side by '5'.
-1.4 + -1.6z + z2 = 0

Move the constant term to the right:

Add '1.4' to each side of the equation.
-1.4 + -1.6z + 1.4 + z2 = 0 + 1.4

Reorder the terms:
-1.4 + 1.4 + -1.6z + z2 = 0 + 1.4

Combine like terms: -1.4 + 1.4 = 0.0
0.0 + -1.6z + z2 = 0 + 1.4
-1.6z + z2 = 0 + 1.4

Combine like terms: 0 + 1.4 = 1.4
-1.6z + z2 = 1.4

The z term is -1.6z.  Take half its coefficient (-0.8).
Square it (0.64) and add it to both sides.

Add '0.64' to each side of the equation.
-1.6z + 0.64 + z2 = 1.4 + 0.64

Reorder the terms:
0.64 + -1.6z + z2 = 1.4 + 0.64

Combine like terms: 1.4 + 0.64 = 2.04
0.64 + -1.6z + z2 = 2.04

Factor a perfect square on the left side:
(z + -0.8)(z + -0.8) = 2.04

Calculate the square root of the right side: 1.428285686

Break this problem into two subproblems by setting 
(z + -0.8) equal to 1.428285686 and -1.428285686.

Subproblem 1

z + -0.8 = 1.428285686 Simplifying z + -0.8 = 1.428285686 Reorder the terms: -0.8 + z = 1.428285686 Solving -0.8 + z = 1.428285686 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + z = 1.428285686 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = 1.428285686 + 0.8 z = 1.428285686 + 0.8 Combine like terms: 1.428285686 + 0.8 = 2.228285686 z = 2.228285686 Simplifying z = 2.228285686

Subproblem 2

z + -0.8 = -1.428285686 Simplifying z + -0.8 = -1.428285686 Reorder the terms: -0.8 + z = -1.428285686 Solving -0.8 + z = -1.428285686 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + z = -1.428285686 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + z = -1.428285686 + 0.8 z = -1.428285686 + 0.8 Combine like terms: -1.428285686 + 0.8 = -0.628285686 z = -0.628285686 Simplifying z = -0.628285686

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.228285686, -0.628285686}

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