(3z^2+5z-1)-(7z^2-7z+8)+(5z^2-6z+1)=0

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Solution for (3z^2+5z-1)-(7z^2-7z+8)+(5z^2-6z+1)=0 equation:


Simplifying
(3z2 + 5z + -1) + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0

Reorder the terms:
(-1 + 5z + 3z2) + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0

Remove parenthesis around (-1 + 5z + 3z2)
-1 + 5z + 3z2 + -1(7z2 + -7z + 8) + (5z2 + -6z + 1) = 0

Reorder the terms:
-1 + 5z + 3z2 + -1(8 + -7z + 7z2) + (5z2 + -6z + 1) = 0
-1 + 5z + 3z2 + (8 * -1 + -7z * -1 + 7z2 * -1) + (5z2 + -6z + 1) = 0
-1 + 5z + 3z2 + (-8 + 7z + -7z2) + (5z2 + -6z + 1) = 0

Reorder the terms:
-1 + 5z + 3z2 + -8 + 7z + -7z2 + (1 + -6z + 5z2) = 0

Remove parenthesis around (1 + -6z + 5z2)
-1 + 5z + 3z2 + -8 + 7z + -7z2 + 1 + -6z + 5z2 = 0

Reorder the terms:
-1 + -8 + 1 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0

Combine like terms: -1 + -8 = -9
-9 + 1 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0

Combine like terms: -9 + 1 = -8
-8 + 5z + 7z + -6z + 3z2 + -7z2 + 5z2 = 0

Combine like terms: 5z + 7z = 12z
-8 + 12z + -6z + 3z2 + -7z2 + 5z2 = 0

Combine like terms: 12z + -6z = 6z
-8 + 6z + 3z2 + -7z2 + 5z2 = 0

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

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

Solving
-8 + 6z + 1z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 6z + 8 + z2 = 0 + 8

Reorder the terms:
-8 + 8 + 6z + z2 = 0 + 8

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

Combine like terms: 0 + 8 = 8
6z + z2 = 8

The z term is 6z.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6z + 9 + z2 = 8 + 9

Reorder the terms:
9 + 6z + z2 = 8 + 9

Combine like terms: 8 + 9 = 17
9 + 6z + z2 = 17

Factor a perfect square on the left side:
(z + 3)(z + 3) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(z + 3) equal to 4.123105626 and -4.123105626.

Subproblem 1

z + 3 = 4.123105626 Simplifying z + 3 = 4.123105626 Reorder the terms: 3 + z = 4.123105626 Solving 3 + z = 4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = 4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = 4.123105626 + -3 z = 4.123105626 + -3 Combine like terms: 4.123105626 + -3 = 1.123105626 z = 1.123105626 Simplifying z = 1.123105626

Subproblem 2

z + 3 = -4.123105626 Simplifying z + 3 = -4.123105626 Reorder the terms: 3 + z = -4.123105626 Solving 3 + z = -4.123105626 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + z = -4.123105626 + -3 Combine like terms: 3 + -3 = 0 0 + z = -4.123105626 + -3 z = -4.123105626 + -3 Combine like terms: -4.123105626 + -3 = -7.123105626 z = -7.123105626 Simplifying z = -7.123105626

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.123105626, -7.123105626}

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