4z^2+33=24z

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Solution for 4z^2+33=24z equation:


Simplifying
4z2 + 33 = 24z

Reorder the terms:
33 + 4z2 = 24z

Solving
33 + 4z2 = 24z

Solving for variable 'z'.

Reorder the terms:
33 + -24z + 4z2 = 24z + -24z

Combine like terms: 24z + -24z = 0
33 + -24z + 4z2 = 0

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

Divide each side by '4'.
8.25 + -6z + z2 = 0

Move the constant term to the right:

Add '-8.25' to each side of the equation.
8.25 + -6z + -8.25 + z2 = 0 + -8.25

Reorder the terms:
8.25 + -8.25 + -6z + z2 = 0 + -8.25

Combine like terms: 8.25 + -8.25 = 0.00
0.00 + -6z + z2 = 0 + -8.25
-6z + z2 = 0 + -8.25

Combine like terms: 0 + -8.25 = -8.25
-6z + z2 = -8.25

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 = -8.25 + 9

Reorder the terms:
9 + -6z + z2 = -8.25 + 9

Combine like terms: -8.25 + 9 = 0.75
9 + -6z + z2 = 0.75

Factor a perfect square on the left side:
(z + -3)(z + -3) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(z + -3) equal to 0.866025404 and -0.866025404.

Subproblem 1

z + -3 = 0.866025404 Simplifying z + -3 = 0.866025404 Reorder the terms: -3 + z = 0.866025404 Solving -3 + z = 0.866025404 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 = 0.866025404 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.866025404 + 3 z = 0.866025404 + 3 Combine like terms: 0.866025404 + 3 = 3.866025404 z = 3.866025404 Simplifying z = 3.866025404

Subproblem 2

z + -3 = -0.866025404 Simplifying z + -3 = -0.866025404 Reorder the terms: -3 + z = -0.866025404 Solving -3 + z = -0.866025404 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 = -0.866025404 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.866025404 + 3 z = -0.866025404 + 3 Combine like terms: -0.866025404 + 3 = 2.133974596 z = 2.133974596 Simplifying z = 2.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.866025404, 2.133974596}

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