7z^2+6z+1=0

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Solution for 7z^2+6z+1=0 equation:


Simplifying
7z2 + 6z + 1 = 0

Reorder the terms:
1 + 6z + 7z2 = 0

Solving
1 + 6z + 7z2 = 0

Solving for variable 'z'.

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

Divide each side by '7'.
0.1428571429 + 0.8571428571z + z2 = 0

Move the constant term to the right:

Add '-0.1428571429' to each side of the equation.
0.1428571429 + 0.8571428571z + -0.1428571429 + z2 = 0 + -0.1428571429

Reorder the terms:
0.1428571429 + -0.1428571429 + 0.8571428571z + z2 = 0 + -0.1428571429

Combine like terms: 0.1428571429 + -0.1428571429 = 0.0000000000
0.0000000000 + 0.8571428571z + z2 = 0 + -0.1428571429
0.8571428571z + z2 = 0 + -0.1428571429

Combine like terms: 0 + -0.1428571429 = -0.1428571429
0.8571428571z + z2 = -0.1428571429

The z term is 0.8571428571z.  Take half its coefficient (0.4285714286).
Square it (0.1836734694) and add it to both sides.

Add '0.1836734694' to each side of the equation.
0.8571428571z + 0.1836734694 + z2 = -0.1428571429 + 0.1836734694

Reorder the terms:
0.1836734694 + 0.8571428571z + z2 = -0.1428571429 + 0.1836734694

Combine like terms: -0.1428571429 + 0.1836734694 = 0.0408163265
0.1836734694 + 0.8571428571z + z2 = 0.0408163265

Factor a perfect square on the left side:
(z + 0.4285714286)(z + 0.4285714286) = 0.0408163265

Calculate the square root of the right side: 0.202030509

Break this problem into two subproblems by setting 
(z + 0.4285714286) equal to 0.202030509 and -0.202030509.

Subproblem 1

z + 0.4285714286 = 0.202030509 Simplifying z + 0.4285714286 = 0.202030509 Reorder the terms: 0.4285714286 + z = 0.202030509 Solving 0.4285714286 + z = 0.202030509 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = 0.202030509 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = 0.202030509 + -0.4285714286 z = 0.202030509 + -0.4285714286 Combine like terms: 0.202030509 + -0.4285714286 = -0.2265409196 z = -0.2265409196 Simplifying z = -0.2265409196

Subproblem 2

z + 0.4285714286 = -0.202030509 Simplifying z + 0.4285714286 = -0.202030509 Reorder the terms: 0.4285714286 + z = -0.202030509 Solving 0.4285714286 + z = -0.202030509 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = -0.202030509 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = -0.202030509 + -0.4285714286 z = -0.202030509 + -0.4285714286 Combine like terms: -0.202030509 + -0.4285714286 = -0.6306019376 z = -0.6306019376 Simplifying z = -0.6306019376

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.2265409196, -0.6306019376}

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