3z^2+12z-49=0

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Solution for 3z^2+12z-49=0 equation:


Simplifying
3z2 + 12z + -49 = 0

Reorder the terms:
-49 + 12z + 3z2 = 0

Solving
-49 + 12z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
-16.33333333 + 4z + z2 = 0

Move the constant term to the right:

Add '16.33333333' to each side of the equation.
-16.33333333 + 4z + 16.33333333 + z2 = 0 + 16.33333333

Reorder the terms:
-16.33333333 + 16.33333333 + 4z + z2 = 0 + 16.33333333

Combine like terms: -16.33333333 + 16.33333333 = 0.00000000
0.00000000 + 4z + z2 = 0 + 16.33333333
4z + z2 = 0 + 16.33333333

Combine like terms: 0 + 16.33333333 = 16.33333333
4z + z2 = 16.33333333

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 = 16.33333333 + 4

Reorder the terms:
4 + 4z + z2 = 16.33333333 + 4

Combine like terms: 16.33333333 + 4 = 20.33333333
4 + 4z + z2 = 20.33333333

Factor a perfect square on the left side:
(z + 2)(z + 2) = 20.33333333

Calculate the square root of the right side: 4.509249752

Break this problem into two subproblems by setting 
(z + 2) equal to 4.509249752 and -4.509249752.

Subproblem 1

z + 2 = 4.509249752 Simplifying z + 2 = 4.509249752 Reorder the terms: 2 + z = 4.509249752 Solving 2 + z = 4.509249752 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 = 4.509249752 + -2 Combine like terms: 2 + -2 = 0 0 + z = 4.509249752 + -2 z = 4.509249752 + -2 Combine like terms: 4.509249752 + -2 = 2.509249752 z = 2.509249752 Simplifying z = 2.509249752

Subproblem 2

z + 2 = -4.509249752 Simplifying z + 2 = -4.509249752 Reorder the terms: 2 + z = -4.509249752 Solving 2 + z = -4.509249752 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 = -4.509249752 + -2 Combine like terms: 2 + -2 = 0 0 + z = -4.509249752 + -2 z = -4.509249752 + -2 Combine like terms: -4.509249752 + -2 = -6.509249752 z = -6.509249752 Simplifying z = -6.509249752

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.509249752, -6.509249752}

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