9z^2-7z-4=0

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Solution for 9z^2-7z-4=0 equation:


Simplifying
9z2 + -7z + -4 = 0

Reorder the terms:
-4 + -7z + 9z2 = 0

Solving
-4 + -7z + 9z2 = 0

Solving for variable 'z'.

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

Divide each side by '9'.
-0.4444444444 + -0.7777777778z + z2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + -0.7777777778z + 0.4444444444 + z2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + -0.7777777778z + z2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + -0.7777777778z + z2 = 0 + 0.4444444444
-0.7777777778z + z2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
-0.7777777778z + z2 = 0.4444444444

The z term is -0.7777777778z.  Take half its coefficient (-0.3888888889).
Square it (0.1512345679) and add it to both sides.

Add '0.1512345679' to each side of the equation.
-0.7777777778z + 0.1512345679 + z2 = 0.4444444444 + 0.1512345679

Reorder the terms:
0.1512345679 + -0.7777777778z + z2 = 0.4444444444 + 0.1512345679

Combine like terms: 0.4444444444 + 0.1512345679 = 0.5956790123
0.1512345679 + -0.7777777778z + z2 = 0.5956790123

Factor a perfect square on the left side:
(z + -0.3888888889)(z + -0.3888888889) = 0.5956790123

Calculate the square root of the right side: 0.771802444

Break this problem into two subproblems by setting 
(z + -0.3888888889) equal to 0.771802444 and -0.771802444.

Subproblem 1

z + -0.3888888889 = 0.771802444 Simplifying z + -0.3888888889 = 0.771802444 Reorder the terms: -0.3888888889 + z = 0.771802444 Solving -0.3888888889 + z = 0.771802444 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3888888889' to each side of the equation. -0.3888888889 + 0.3888888889 + z = 0.771802444 + 0.3888888889 Combine like terms: -0.3888888889 + 0.3888888889 = 0.0000000000 0.0000000000 + z = 0.771802444 + 0.3888888889 z = 0.771802444 + 0.3888888889 Combine like terms: 0.771802444 + 0.3888888889 = 1.1606913329 z = 1.1606913329 Simplifying z = 1.1606913329

Subproblem 2

z + -0.3888888889 = -0.771802444 Simplifying z + -0.3888888889 = -0.771802444 Reorder the terms: -0.3888888889 + z = -0.771802444 Solving -0.3888888889 + z = -0.771802444 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.3888888889' to each side of the equation. -0.3888888889 + 0.3888888889 + z = -0.771802444 + 0.3888888889 Combine like terms: -0.3888888889 + 0.3888888889 = 0.0000000000 0.0000000000 + z = -0.771802444 + 0.3888888889 z = -0.771802444 + 0.3888888889 Combine like terms: -0.771802444 + 0.3888888889 = -0.3829135551 z = -0.3829135551 Simplifying z = -0.3829135551

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.1606913329, -0.3829135551}

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