3z^2+13z=21

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Solution for 3z^2+13z=21 equation:


Simplifying
3z2 + 13z = 21

Reorder the terms:
13z + 3z2 = 21

Solving
13z + 3z2 = 21

Solving for variable 'z'.

Reorder the terms:
-21 + 13z + 3z2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 13z + 3z2 = 0

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

Divide each side by '3'.
-7 + 4.333333333z + z2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 4.333333333z + 7 + z2 = 0 + 7

Reorder the terms:
-7 + 7 + 4.333333333z + z2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 4.333333333z + z2 = 0 + 7
4.333333333z + z2 = 0 + 7

Combine like terms: 0 + 7 = 7
4.333333333z + z2 = 7

The z term is 4.333333333z.  Take half its coefficient (2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
4.333333333z + 4.694444446 + z2 = 7 + 4.694444446

Reorder the terms:
4.694444446 + 4.333333333z + z2 = 7 + 4.694444446

Combine like terms: 7 + 4.694444446 = 11.694444446
4.694444446 + 4.333333333z + z2 = 11.694444446

Factor a perfect square on the left side:
(z + 2.166666667)(z + 2.166666667) = 11.694444446

Calculate the square root of the right side: 3.419714088

Break this problem into two subproblems by setting 
(z + 2.166666667) equal to 3.419714088 and -3.419714088.

Subproblem 1

z + 2.166666667 = 3.419714088 Simplifying z + 2.166666667 = 3.419714088 Reorder the terms: 2.166666667 + z = 3.419714088 Solving 2.166666667 + z = 3.419714088 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + z = 3.419714088 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + z = 3.419714088 + -2.166666667 z = 3.419714088 + -2.166666667 Combine like terms: 3.419714088 + -2.166666667 = 1.253047421 z = 1.253047421 Simplifying z = 1.253047421

Subproblem 2

z + 2.166666667 = -3.419714088 Simplifying z + 2.166666667 = -3.419714088 Reorder the terms: 2.166666667 + z = -3.419714088 Solving 2.166666667 + z = -3.419714088 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.166666667' to each side of the equation. 2.166666667 + -2.166666667 + z = -3.419714088 + -2.166666667 Combine like terms: 2.166666667 + -2.166666667 = 0.000000000 0.000000000 + z = -3.419714088 + -2.166666667 z = -3.419714088 + -2.166666667 Combine like terms: -3.419714088 + -2.166666667 = -5.586380755 z = -5.586380755 Simplifying z = -5.586380755

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.253047421, -5.586380755}

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