9z^2-54z+76=

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Solution for 9z^2-54z+76= equation:


Simplifying
9z2 + -54z + 76 = 0

Reorder the terms:
76 + -54z + 9z2 = 0

Solving
76 + -54z + 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'.
8.444444444 + -6z + z2 = 0

Move the constant term to the right:

Add '-8.444444444' to each side of the equation.
8.444444444 + -6z + -8.444444444 + z2 = 0 + -8.444444444

Reorder the terms:
8.444444444 + -8.444444444 + -6z + z2 = 0 + -8.444444444

Combine like terms: 8.444444444 + -8.444444444 = 0.000000000
0.000000000 + -6z + z2 = 0 + -8.444444444
-6z + z2 = 0 + -8.444444444

Combine like terms: 0 + -8.444444444 = -8.444444444
-6z + z2 = -8.444444444

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = -8.444444444 + 9

Reorder the terms:
9 + -6z + z2 = -8.444444444 + 9

Combine like terms: -8.444444444 + 9 = 0.555555556
9 + -6z + z2 = 0.555555556

Factor a perfect square on the left side:
(z + -3)(z + -3) = 0.555555556

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(z + -3) equal to 0.745355993 and -0.745355993.

Subproblem 1

z + -3 = 0.745355993 Simplifying z + -3 = 0.745355993 Reorder the terms: -3 + z = 0.745355993 Solving -3 + z = 0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 0.745355993 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.745355993 + 3 z = 0.745355993 + 3 Combine like terms: 0.745355993 + 3 = 3.745355993 z = 3.745355993 Simplifying z = 3.745355993

Subproblem 2

z + -3 = -0.745355993 Simplifying z + -3 = -0.745355993 Reorder the terms: -3 + z = -0.745355993 Solving -3 + z = -0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -0.745355993 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.745355993 + 3 z = -0.745355993 + 3 Combine like terms: -0.745355993 + 3 = 2.254644007 z = 2.254644007 Simplifying z = 2.254644007

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.745355993, 2.254644007}

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