9z^2+10z-4=0

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Solution for 9z^2+10z-4=0 equation:


Simplifying
9z2 + 10z + -4 = 0

Reorder the terms:
-4 + 10z + 9z2 = 0

Solving
-4 + 10z + 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 + 1.111111111z + z2 = 0

Move the constant term to the right:

Add '0.4444444444' to each side of the equation.
-0.4444444444 + 1.111111111z + 0.4444444444 + z2 = 0 + 0.4444444444

Reorder the terms:
-0.4444444444 + 0.4444444444 + 1.111111111z + z2 = 0 + 0.4444444444

Combine like terms: -0.4444444444 + 0.4444444444 = 0.0000000000
0.0000000000 + 1.111111111z + z2 = 0 + 0.4444444444
1.111111111z + z2 = 0 + 0.4444444444

Combine like terms: 0 + 0.4444444444 = 0.4444444444
1.111111111z + z2 = 0.4444444444

The z term is 1.111111111z.  Take half its coefficient (0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
1.111111111z + 0.3086419752 + z2 = 0.4444444444 + 0.3086419752

Reorder the terms:
0.3086419752 + 1.111111111z + z2 = 0.4444444444 + 0.3086419752

Combine like terms: 0.4444444444 + 0.3086419752 = 0.7530864196
0.3086419752 + 1.111111111z + z2 = 0.7530864196

Factor a perfect square on the left side:
(z + 0.5555555555)(z + 0.5555555555) = 0.7530864196

Calculate the square root of the right side: 0.867805519

Break this problem into two subproblems by setting 
(z + 0.5555555555) equal to 0.867805519 and -0.867805519.

Subproblem 1

z + 0.5555555555 = 0.867805519 Simplifying z + 0.5555555555 = 0.867805519 Reorder the terms: 0.5555555555 + z = 0.867805519 Solving 0.5555555555 + z = 0.867805519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + z = 0.867805519 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + z = 0.867805519 + -0.5555555555 z = 0.867805519 + -0.5555555555 Combine like terms: 0.867805519 + -0.5555555555 = 0.3122499635 z = 0.3122499635 Simplifying z = 0.3122499635

Subproblem 2

z + 0.5555555555 = -0.867805519 Simplifying z + 0.5555555555 = -0.867805519 Reorder the terms: 0.5555555555 + z = -0.867805519 Solving 0.5555555555 + z = -0.867805519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + z = -0.867805519 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + z = -0.867805519 + -0.5555555555 z = -0.867805519 + -0.5555555555 Combine like terms: -0.867805519 + -0.5555555555 = -1.4233610745 z = -1.4233610745 Simplifying z = -1.4233610745

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.3122499635, -1.4233610745}

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