7z^2+8z-5=0

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Solution for 7z^2+8z-5=0 equation:


Simplifying
7z2 + 8z + -5 = 0

Reorder the terms:
-5 + 8z + 7z2 = 0

Solving
-5 + 8z + 7z2 = 0

Solving for variable 'z'.

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

Divide each side by '7'.
-0.7142857143 + 1.142857143z + z2 = 0

Move the constant term to the right:

Add '0.7142857143' to each side of the equation.
-0.7142857143 + 1.142857143z + 0.7142857143 + z2 = 0 + 0.7142857143

Reorder the terms:
-0.7142857143 + 0.7142857143 + 1.142857143z + z2 = 0 + 0.7142857143

Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000
0.0000000000 + 1.142857143z + z2 = 0 + 0.7142857143
1.142857143z + z2 = 0 + 0.7142857143

Combine like terms: 0 + 0.7142857143 = 0.7142857143
1.142857143z + z2 = 0.7142857143

The z term is 1.142857143z.  Take half its coefficient (0.5714285715).
Square it (0.3265306123) and add it to both sides.

Add '0.3265306123' to each side of the equation.
1.142857143z + 0.3265306123 + z2 = 0.7142857143 + 0.3265306123

Reorder the terms:
0.3265306123 + 1.142857143z + z2 = 0.7142857143 + 0.3265306123

Combine like terms: 0.7142857143 + 0.3265306123 = 1.0408163266
0.3265306123 + 1.142857143z + z2 = 1.0408163266

Factor a perfect square on the left side:
(z + 0.5714285715)(z + 0.5714285715) = 1.0408163266

Calculate the square root of the right side: 1.020204061

Break this problem into two subproblems by setting 
(z + 0.5714285715) equal to 1.020204061 and -1.020204061.

Subproblem 1

z + 0.5714285715 = 1.020204061 Simplifying z + 0.5714285715 = 1.020204061 Reorder the terms: 0.5714285715 + z = 1.020204061 Solving 0.5714285715 + z = 1.020204061 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + z = 1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + z = 1.020204061 + -0.5714285715 z = 1.020204061 + -0.5714285715 Combine like terms: 1.020204061 + -0.5714285715 = 0.4487754895 z = 0.4487754895 Simplifying z = 0.4487754895

Subproblem 2

z + 0.5714285715 = -1.020204061 Simplifying z + 0.5714285715 = -1.020204061 Reorder the terms: 0.5714285715 + z = -1.020204061 Solving 0.5714285715 + z = -1.020204061 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + z = -1.020204061 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + z = -1.020204061 + -0.5714285715 z = -1.020204061 + -0.5714285715 Combine like terms: -1.020204061 + -0.5714285715 = -1.5916326325 z = -1.5916326325 Simplifying z = -1.5916326325

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.4487754895, -1.5916326325}

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