6z^2+17z-56=0

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Solution for 6z^2+17z-56=0 equation:


Simplifying
6z2 + 17z + -56 = 0

Reorder the terms:
-56 + 17z + 6z2 = 0

Solving
-56 + 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'.
-9.333333333 + 2.833333333z + z2 = 0

Move the constant term to the right:

Add '9.333333333' to each side of the equation.
-9.333333333 + 2.833333333z + 9.333333333 + z2 = 0 + 9.333333333

Reorder the terms:
-9.333333333 + 9.333333333 + 2.833333333z + z2 = 0 + 9.333333333

Combine like terms: -9.333333333 + 9.333333333 = 0.000000000
0.000000000 + 2.833333333z + z2 = 0 + 9.333333333
2.833333333z + z2 = 0 + 9.333333333

Combine like terms: 0 + 9.333333333 = 9.333333333
2.833333333z + z2 = 9.333333333

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 = 9.333333333 + 2.006944445

Reorder the terms:
2.006944445 + 2.833333333z + z2 = 9.333333333 + 2.006944445

Combine like terms: 9.333333333 + 2.006944445 = 11.340277778
2.006944445 + 2.833333333z + z2 = 11.340277778

Factor a perfect square on the left side:
(z + 1.416666667)(z + 1.416666667) = 11.340277778

Calculate the square root of the right side: 3.367532892

Break this problem into two subproblems by setting 
(z + 1.416666667) equal to 3.367532892 and -3.367532892.

Subproblem 1

z + 1.416666667 = 3.367532892 Simplifying z + 1.416666667 = 3.367532892 Reorder the terms: 1.416666667 + z = 3.367532892 Solving 1.416666667 + z = 3.367532892 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 = 3.367532892 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = 3.367532892 + -1.416666667 z = 3.367532892 + -1.416666667 Combine like terms: 3.367532892 + -1.416666667 = 1.950866225 z = 1.950866225 Simplifying z = 1.950866225

Subproblem 2

z + 1.416666667 = -3.367532892 Simplifying z + 1.416666667 = -3.367532892 Reorder the terms: 1.416666667 + z = -3.367532892 Solving 1.416666667 + z = -3.367532892 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 = -3.367532892 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + z = -3.367532892 + -1.416666667 z = -3.367532892 + -1.416666667 Combine like terms: -3.367532892 + -1.416666667 = -4.784199559 z = -4.784199559 Simplifying z = -4.784199559

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.950866225, -4.784199559}

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