5z^2-9z-7=0

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Solution for 5z^2-9z-7=0 equation:


Simplifying
5z2 + -9z + -7 = 0

Reorder the terms:
-7 + -9z + 5z2 = 0

Solving
-7 + -9z + 5z2 = 0

Solving for variable 'z'.

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

Divide each side by '5'.
-1.4 + -1.8z + z2 = 0

Move the constant term to the right:

Add '1.4' to each side of the equation.
-1.4 + -1.8z + 1.4 + z2 = 0 + 1.4

Reorder the terms:
-1.4 + 1.4 + -1.8z + z2 = 0 + 1.4

Combine like terms: -1.4 + 1.4 = 0.0
0.0 + -1.8z + z2 = 0 + 1.4
-1.8z + z2 = 0 + 1.4

Combine like terms: 0 + 1.4 = 1.4
-1.8z + z2 = 1.4

The z term is -1.8z.  Take half its coefficient (-0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
-1.8z + 0.81 + z2 = 1.4 + 0.81

Reorder the terms:
0.81 + -1.8z + z2 = 1.4 + 0.81

Combine like terms: 1.4 + 0.81 = 2.21
0.81 + -1.8z + z2 = 2.21

Factor a perfect square on the left side:
(z + -0.9)(z + -0.9) = 2.21

Calculate the square root of the right side: 1.486606875

Break this problem into two subproblems by setting 
(z + -0.9) equal to 1.486606875 and -1.486606875.

Subproblem 1

z + -0.9 = 1.486606875 Simplifying z + -0.9 = 1.486606875 Reorder the terms: -0.9 + z = 1.486606875 Solving -0.9 + z = 1.486606875 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + z = 1.486606875 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + z = 1.486606875 + 0.9 z = 1.486606875 + 0.9 Combine like terms: 1.486606875 + 0.9 = 2.386606875 z = 2.386606875 Simplifying z = 2.386606875

Subproblem 2

z + -0.9 = -1.486606875 Simplifying z + -0.9 = -1.486606875 Reorder the terms: -0.9 + z = -1.486606875 Solving -0.9 + z = -1.486606875 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.9' to each side of the equation. -0.9 + 0.9 + z = -1.486606875 + 0.9 Combine like terms: -0.9 + 0.9 = 0.0 0.0 + z = -1.486606875 + 0.9 z = -1.486606875 + 0.9 Combine like terms: -1.486606875 + 0.9 = -0.586606875 z = -0.586606875 Simplifying z = -0.586606875

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.386606875, -0.586606875}

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