6z^2-15z-3=-17z

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Solution for 6z^2-15z-3=-17z equation:


Simplifying
6z2 + -15z + -3 = -17z

Reorder the terms:
-3 + -15z + 6z2 = -17z

Solving
-3 + -15z + 6z2 = -17z

Solving for variable 'z'.

Reorder the terms:
-3 + -15z + 17z + 6z2 = -17z + 17z

Combine like terms: -15z + 17z = 2z
-3 + 2z + 6z2 = -17z + 17z

Combine like terms: -17z + 17z = 0
-3 + 2z + 6z2 = 0

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

Divide each side by '6'.
-0.5 + 0.3333333333z + z2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 0.3333333333z + 0.5 + z2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 0.3333333333z + z2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 0.3333333333z + z2 = 0 + 0.5
0.3333333333z + z2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
0.3333333333z + z2 = 0.5

The z term is 0.3333333333z.  Take half its coefficient (0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
0.3333333333z + 0.02777777779 + z2 = 0.5 + 0.02777777779

Reorder the terms:
0.02777777779 + 0.3333333333z + z2 = 0.5 + 0.02777777779

Combine like terms: 0.5 + 0.02777777779 = 0.52777777779
0.02777777779 + 0.3333333333z + z2 = 0.52777777779

Factor a perfect square on the left side:
(z + 0.1666666667)(z + 0.1666666667) = 0.52777777779

Calculate the square root of the right side: 0.726483157

Break this problem into two subproblems by setting 
(z + 0.1666666667) equal to 0.726483157 and -0.726483157.

Subproblem 1

z + 0.1666666667 = 0.726483157 Simplifying z + 0.1666666667 = 0.726483157 Reorder the terms: 0.1666666667 + z = 0.726483157 Solving 0.1666666667 + z = 0.726483157 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = 0.726483157 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = 0.726483157 + -0.1666666667 z = 0.726483157 + -0.1666666667 Combine like terms: 0.726483157 + -0.1666666667 = 0.5598164903 z = 0.5598164903 Simplifying z = 0.5598164903

Subproblem 2

z + 0.1666666667 = -0.726483157 Simplifying z + 0.1666666667 = -0.726483157 Reorder the terms: 0.1666666667 + z = -0.726483157 Solving 0.1666666667 + z = -0.726483157 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + z = -0.726483157 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + z = -0.726483157 + -0.1666666667 z = -0.726483157 + -0.1666666667 Combine like terms: -0.726483157 + -0.1666666667 = -0.8931498237 z = -0.8931498237 Simplifying z = -0.8931498237

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.5598164903, -0.8931498237}

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