21z^2+85z-28=0

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Solution for 21z^2+85z-28=0 equation:


Simplifying
21z2 + 85z + -28 = 0

Reorder the terms:
-28 + 85z + 21z2 = 0

Solving
-28 + 85z + 21z2 = 0

Solving for variable 'z'.

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

Divide each side by '21'.
-1.333333333 + 4.047619048z + z2 = 0

Move the constant term to the right:

Add '1.333333333' to each side of the equation.
-1.333333333 + 4.047619048z + 1.333333333 + z2 = 0 + 1.333333333

Reorder the terms:
-1.333333333 + 1.333333333 + 4.047619048z + z2 = 0 + 1.333333333

Combine like terms: -1.333333333 + 1.333333333 = 0.000000000
0.000000000 + 4.047619048z + z2 = 0 + 1.333333333
4.047619048z + z2 = 0 + 1.333333333

Combine like terms: 0 + 1.333333333 = 1.333333333
4.047619048z + z2 = 1.333333333

The z term is 4.047619048z.  Take half its coefficient (2.023809524).
Square it (4.095804989) and add it to both sides.

Add '4.095804989' to each side of the equation.
4.047619048z + 4.095804989 + z2 = 1.333333333 + 4.095804989

Reorder the terms:
4.095804989 + 4.047619048z + z2 = 1.333333333 + 4.095804989

Combine like terms: 1.333333333 + 4.095804989 = 5.429138322
4.095804989 + 4.047619048z + z2 = 5.429138322

Factor a perfect square on the left side:
(z + 2.023809524)(z + 2.023809524) = 5.429138322

Calculate the square root of the right side: 2.330051141

Break this problem into two subproblems by setting 
(z + 2.023809524) equal to 2.330051141 and -2.330051141.

Subproblem 1

z + 2.023809524 = 2.330051141 Simplifying z + 2.023809524 = 2.330051141 Reorder the terms: 2.023809524 + z = 2.330051141 Solving 2.023809524 + z = 2.330051141 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.023809524' to each side of the equation. 2.023809524 + -2.023809524 + z = 2.330051141 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = 2.330051141 + -2.023809524 z = 2.330051141 + -2.023809524 Combine like terms: 2.330051141 + -2.023809524 = 0.306241617 z = 0.306241617 Simplifying z = 0.306241617

Subproblem 2

z + 2.023809524 = -2.330051141 Simplifying z + 2.023809524 = -2.330051141 Reorder the terms: 2.023809524 + z = -2.330051141 Solving 2.023809524 + z = -2.330051141 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.023809524' to each side of the equation. 2.023809524 + -2.023809524 + z = -2.330051141 + -2.023809524 Combine like terms: 2.023809524 + -2.023809524 = 0.000000000 0.000000000 + z = -2.330051141 + -2.023809524 z = -2.330051141 + -2.023809524 Combine like terms: -2.330051141 + -2.023809524 = -4.353860665 z = -4.353860665 Simplifying z = -4.353860665

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.306241617, -4.353860665}

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