9z^2+31=36z

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Solution for 9z^2+31=36z equation:


Simplifying
9z2 + 31 = 36z

Reorder the terms:
31 + 9z2 = 36z

Solving
31 + 9z2 = 36z

Solving for variable 'z'.

Reorder the terms:
31 + -36z + 9z2 = 36z + -36z

Combine like terms: 36z + -36z = 0
31 + -36z + 9z2 = 0

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

Divide each side by '9'.
3.444444444 + -4z + z2 = 0

Move the constant term to the right:

Add '-3.444444444' to each side of the equation.
3.444444444 + -4z + -3.444444444 + z2 = 0 + -3.444444444

Reorder the terms:
3.444444444 + -3.444444444 + -4z + z2 = 0 + -3.444444444

Combine like terms: 3.444444444 + -3.444444444 = 0.000000000
0.000000000 + -4z + z2 = 0 + -3.444444444
-4z + z2 = 0 + -3.444444444

Combine like terms: 0 + -3.444444444 = -3.444444444
-4z + z2 = -3.444444444

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = -3.444444444 + 4

Reorder the terms:
4 + -4z + z2 = -3.444444444 + 4

Combine like terms: -3.444444444 + 4 = 0.555555556
4 + -4z + z2 = 0.555555556

Factor a perfect square on the left side:
(z + -2)(z + -2) = 0.555555556

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(z + -2) equal to 0.745355993 and -0.745355993.

Subproblem 1

z + -2 = 0.745355993 Simplifying z + -2 = 0.745355993 Reorder the terms: -2 + z = 0.745355993 Solving -2 + z = 0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.745355993 + 2 z = 0.745355993 + 2 Combine like terms: 0.745355993 + 2 = 2.745355993 z = 2.745355993 Simplifying z = 2.745355993

Subproblem 2

z + -2 = -0.745355993 Simplifying z + -2 = -0.745355993 Reorder the terms: -2 + z = -0.745355993 Solving -2 + z = -0.745355993 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -0.745355993 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.745355993 + 2 z = -0.745355993 + 2 Combine like terms: -0.745355993 + 2 = 1.254644007 z = 1.254644007 Simplifying z = 1.254644007

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.745355993, 1.254644007}

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