(6z^2-5z+1)+(8z^2+7z-4)=0

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Solution for (6z^2-5z+1)+(8z^2+7z-4)=0 equation:


Simplifying
(6z2 + -5z + 1) + (8z2 + 7z + -4) = 0

Reorder the terms:
(1 + -5z + 6z2) + (8z2 + 7z + -4) = 0

Remove parenthesis around (1 + -5z + 6z2)
1 + -5z + 6z2 + (8z2 + 7z + -4) = 0

Reorder the terms:
1 + -5z + 6z2 + (-4 + 7z + 8z2) = 0

Remove parenthesis around (-4 + 7z + 8z2)
1 + -5z + 6z2 + -4 + 7z + 8z2 = 0

Reorder the terms:
1 + -4 + -5z + 7z + 6z2 + 8z2 = 0

Combine like terms: 1 + -4 = -3
-3 + -5z + 7z + 6z2 + 8z2 = 0

Combine like terms: -5z + 7z = 2z
-3 + 2z + 6z2 + 8z2 = 0

Combine like terms: 6z2 + 8z2 = 14z2
-3 + 2z + 14z2 = 0

Solving
-3 + 2z + 14z2 = 0

Solving for variable 'z'.

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

Divide each side by '14'.
-0.2142857143 + 0.1428571429z + z2 = 0

Move the constant term to the right:

Add '0.2142857143' to each side of the equation.
-0.2142857143 + 0.1428571429z + 0.2142857143 + z2 = 0 + 0.2142857143

Reorder the terms:
-0.2142857143 + 0.2142857143 + 0.1428571429z + z2 = 0 + 0.2142857143

Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000
0.0000000000 + 0.1428571429z + z2 = 0 + 0.2142857143
0.1428571429z + z2 = 0 + 0.2142857143

Combine like terms: 0 + 0.2142857143 = 0.2142857143
0.1428571429z + z2 = 0.2142857143

The z term is 0.1428571429z.  Take half its coefficient (0.07142857145).
Square it (0.005102040819) and add it to both sides.

Add '0.005102040819' to each side of the equation.
0.1428571429z + 0.005102040819 + z2 = 0.2142857143 + 0.005102040819

Reorder the terms:
0.005102040819 + 0.1428571429z + z2 = 0.2142857143 + 0.005102040819

Combine like terms: 0.2142857143 + 0.005102040819 = 0.219387755119
0.005102040819 + 0.1428571429z + z2 = 0.219387755119

Factor a perfect square on the left side:
(z + 0.07142857145)(z + 0.07142857145) = 0.219387755119

Calculate the square root of the right side: 0.468388466

Break this problem into two subproblems by setting 
(z + 0.07142857145) equal to 0.468388466 and -0.468388466.

Subproblem 1

z + 0.07142857145 = 0.468388466 Simplifying z + 0.07142857145 = 0.468388466 Reorder the terms: 0.07142857145 + z = 0.468388466 Solving 0.07142857145 + z = 0.468388466 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + z = 0.468388466 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + z = 0.468388466 + -0.07142857145 z = 0.468388466 + -0.07142857145 Combine like terms: 0.468388466 + -0.07142857145 = 0.39695989455 z = 0.39695989455 Simplifying z = 0.39695989455

Subproblem 2

z + 0.07142857145 = -0.468388466 Simplifying z + 0.07142857145 = -0.468388466 Reorder the terms: 0.07142857145 + z = -0.468388466 Solving 0.07142857145 + z = -0.468388466 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.07142857145' to each side of the equation. 0.07142857145 + -0.07142857145 + z = -0.468388466 + -0.07142857145 Combine like terms: 0.07142857145 + -0.07142857145 = 0.00000000000 0.00000000000 + z = -0.468388466 + -0.07142857145 z = -0.468388466 + -0.07142857145 Combine like terms: -0.468388466 + -0.07142857145 = -0.53981703745 z = -0.53981703745 Simplifying z = -0.53981703745

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.39695989455, -0.53981703745}

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