3z^2+45=28z

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Solution for 3z^2+45=28z equation:


Simplifying
3z2 + 45 = 28z

Reorder the terms:
45 + 3z2 = 28z

Solving
45 + 3z2 = 28z

Solving for variable 'z'.

Reorder the terms:
45 + -28z + 3z2 = 28z + -28z

Combine like terms: 28z + -28z = 0
45 + -28z + 3z2 = 0

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

Divide each side by '3'.
15 + -9.333333333z + z2 = 0

Move the constant term to the right:

Add '-15' to each side of the equation.
15 + -9.333333333z + -15 + z2 = 0 + -15

Reorder the terms:
15 + -15 + -9.333333333z + z2 = 0 + -15

Combine like terms: 15 + -15 = 0
0 + -9.333333333z + z2 = 0 + -15
-9.333333333z + z2 = 0 + -15

Combine like terms: 0 + -15 = -15
-9.333333333z + z2 = -15

The z term is -9.333333333z.  Take half its coefficient (-4.666666667).
Square it (21.77777778) and add it to both sides.

Add '21.77777778' to each side of the equation.
-9.333333333z + 21.77777778 + z2 = -15 + 21.77777778

Reorder the terms:
21.77777778 + -9.333333333z + z2 = -15 + 21.77777778

Combine like terms: -15 + 21.77777778 = 6.77777778
21.77777778 + -9.333333333z + z2 = 6.77777778

Factor a perfect square on the left side:
(z + -4.666666667)(z + -4.666666667) = 6.77777778

Calculate the square root of the right side: 2.603416559

Break this problem into two subproblems by setting 
(z + -4.666666667) equal to 2.603416559 and -2.603416559.

Subproblem 1

z + -4.666666667 = 2.603416559 Simplifying z + -4.666666667 = 2.603416559 Reorder the terms: -4.666666667 + z = 2.603416559 Solving -4.666666667 + z = 2.603416559 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.666666667' to each side of the equation. -4.666666667 + 4.666666667 + z = 2.603416559 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + z = 2.603416559 + 4.666666667 z = 2.603416559 + 4.666666667 Combine like terms: 2.603416559 + 4.666666667 = 7.270083226 z = 7.270083226 Simplifying z = 7.270083226

Subproblem 2

z + -4.666666667 = -2.603416559 Simplifying z + -4.666666667 = -2.603416559 Reorder the terms: -4.666666667 + z = -2.603416559 Solving -4.666666667 + z = -2.603416559 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.666666667' to each side of the equation. -4.666666667 + 4.666666667 + z = -2.603416559 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + z = -2.603416559 + 4.666666667 z = -2.603416559 + 4.666666667 Combine like terms: -2.603416559 + 4.666666667 = 2.063250108 z = 2.063250108 Simplifying z = 2.063250108

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.270083226, 2.063250108}

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