25z^2-30z-9=0

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Solution for 25z^2-30z-9=0 equation:


Simplifying
25z2 + -30z + -9 = 0

Reorder the terms:
-9 + -30z + 25z2 = 0

Solving
-9 + -30z + 25z2 = 0

Solving for variable 'z'.

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

Divide each side by '25'.
-0.36 + -1.2z + z2 = 0

Move the constant term to the right:

Add '0.36' to each side of the equation.
-0.36 + -1.2z + 0.36 + z2 = 0 + 0.36

Reorder the terms:
-0.36 + 0.36 + -1.2z + z2 = 0 + 0.36

Combine like terms: -0.36 + 0.36 = 0.00
0.00 + -1.2z + z2 = 0 + 0.36
-1.2z + z2 = 0 + 0.36

Combine like terms: 0 + 0.36 = 0.36
-1.2z + z2 = 0.36

The z term is -1.2z.  Take half its coefficient (-0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
-1.2z + 0.36 + z2 = 0.36 + 0.36

Reorder the terms:
0.36 + -1.2z + z2 = 0.36 + 0.36

Combine like terms: 0.36 + 0.36 = 0.72
0.36 + -1.2z + z2 = 0.72

Factor a perfect square on the left side:
(z + -0.6)(z + -0.6) = 0.72

Calculate the square root of the right side: 0.848528137

Break this problem into two subproblems by setting 
(z + -0.6) equal to 0.848528137 and -0.848528137.

Subproblem 1

z + -0.6 = 0.848528137 Simplifying z + -0.6 = 0.848528137 Reorder the terms: -0.6 + z = 0.848528137 Solving -0.6 + z = 0.848528137 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6' to each side of the equation. -0.6 + 0.6 + z = 0.848528137 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + z = 0.848528137 + 0.6 z = 0.848528137 + 0.6 Combine like terms: 0.848528137 + 0.6 = 1.448528137 z = 1.448528137 Simplifying z = 1.448528137

Subproblem 2

z + -0.6 = -0.848528137 Simplifying z + -0.6 = -0.848528137 Reorder the terms: -0.6 + z = -0.848528137 Solving -0.6 + z = -0.848528137 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.6' to each side of the equation. -0.6 + 0.6 + z = -0.848528137 + 0.6 Combine like terms: -0.6 + 0.6 = 0.0 0.0 + z = -0.848528137 + 0.6 z = -0.848528137 + 0.6 Combine like terms: -0.848528137 + 0.6 = -0.248528137 z = -0.248528137 Simplifying z = -0.248528137

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.448528137, -0.248528137}

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