6z^2+17z-12=0

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Solution for 6z^2+17z-12=0 equation:


Simplifying
6z2 + 17z + -12 = 0

Reorder the terms:
-12 + 17z + 6z2 = 0

Solving
-12 + 17z + 6z2 = 0

Solving for variable 'z'.

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

Divide each side by '6'.
-2 + 2.833333333z + z2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 2.833333333z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + 2.833333333z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 2.833333333z + z2 = 0 + 2
2.833333333z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
2.833333333z + z2 = 2

The z term is 2.833333333z.  Take half its coefficient (1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
2.833333333z + 2.006944445 + z2 = 2 + 2.006944445

Reorder the terms:
2.006944445 + 2.833333333z + z2 = 2 + 2.006944445

Combine like terms: 2 + 2.006944445 = 4.006944445
2.006944445 + 2.833333333z + z2 = 4.006944445

Factor a perfect square on the left side:
(z + 1.416666667)(z + 1.416666667) = 4.006944445

Calculate the square root of the right side: 2.001735358

Break this problem into two subproblems by setting 
(z + 1.416666667) equal to 2.001735358 and -2.001735358.

Subproblem 1

z + 1.416666667 = 2.001735358 Simplifying z + 1.416666667 = 2.001735358 Reorder the terms: 1.416666667 + z = 2.001735358 Solving 1.416666667 + z = 2.001735358 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + z = 2.001735358 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = 2.001735358 + -1.416666667 z = 2.001735358 + -1.416666667 Combine like terms: 2.001735358 + -1.416666667 = 0.585068691 z = 0.585068691 Simplifying z = 0.585068691

Subproblem 2

z + 1.416666667 = -2.001735358 Simplifying z + 1.416666667 = -2.001735358 Reorder the terms: 1.416666667 + z = -2.001735358 Solving 1.416666667 + z = -2.001735358 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + z = -2.001735358 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = -2.001735358 + -1.416666667 z = -2.001735358 + -1.416666667 Combine like terms: -2.001735358 + -1.416666667 = -3.418402025 z = -3.418402025 Simplifying z = -3.418402025

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.585068691, -3.418402025}

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