0=3z^2-10z-21

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Solution for 0=3z^2-10z-21 equation:


Simplifying
0 = 3z2 + -10z + -21

Reorder the terms:
0 = -21 + -10z + 3z2

Solving
0 = -21 + -10z + 3z2

Solving for variable 'z'.

Combine like terms: 0 + 21 = 21
21 + 10z + -3z2 = -21 + -10z + 3z2 + 21 + 10z + -3z2

Reorder the terms:
21 + 10z + -3z2 = -21 + 21 + -10z + 10z + 3z2 + -3z2

Combine like terms: -21 + 21 = 0
21 + 10z + -3z2 = 0 + -10z + 10z + 3z2 + -3z2
21 + 10z + -3z2 = -10z + 10z + 3z2 + -3z2

Combine like terms: -10z + 10z = 0
21 + 10z + -3z2 = 0 + 3z2 + -3z2
21 + 10z + -3z2 = 3z2 + -3z2

Combine like terms: 3z2 + -3z2 = 0
21 + 10z + -3z2 = 0

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

Divide each side by '-3'.
-7 + -3.333333333z + z2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -3.333333333z + 7 + z2 = 0 + 7

Reorder the terms:
-7 + 7 + -3.333333333z + z2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -3.333333333z + z2 = 0 + 7
-3.333333333z + z2 = 0 + 7

Combine like terms: 0 + 7 = 7
-3.333333333z + z2 = 7

The z term is -3.333333333z.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333z + 2.777777779 + z2 = 7 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333z + z2 = 7 + 2.777777779

Combine like terms: 7 + 2.777777779 = 9.777777779
2.777777779 + -3.333333333z + z2 = 9.777777779

Factor a perfect square on the left side:
(z + -1.666666667)(z + -1.666666667) = 9.777777779

Calculate the square root of the right side: 3.12694384

Break this problem into two subproblems by setting 
(z + -1.666666667) equal to 3.12694384 and -3.12694384.

Subproblem 1

z + -1.666666667 = 3.12694384 Simplifying z + -1.666666667 = 3.12694384 Reorder the terms: -1.666666667 + z = 3.12694384 Solving -1.666666667 + z = 3.12694384 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + z = 3.12694384 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + z = 3.12694384 + 1.666666667 z = 3.12694384 + 1.666666667 Combine like terms: 3.12694384 + 1.666666667 = 4.793610507 z = 4.793610507 Simplifying z = 4.793610507

Subproblem 2

z + -1.666666667 = -3.12694384 Simplifying z + -1.666666667 = -3.12694384 Reorder the terms: -1.666666667 + z = -3.12694384 Solving -1.666666667 + z = -3.12694384 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + z = -3.12694384 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + z = -3.12694384 + 1.666666667 z = -3.12694384 + 1.666666667 Combine like terms: -3.12694384 + 1.666666667 = -1.460277173 z = -1.460277173 Simplifying z = -1.460277173

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.793610507, -1.460277173}

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