4z^2-56z+19=0

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Solution for 4z^2-56z+19=0 equation:


Simplifying
4z2 + -56z + 19 = 0

Reorder the terms:
19 + -56z + 4z2 = 0

Solving
19 + -56z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
4.75 + -14z + z2 = 0

Move the constant term to the right:

Add '-4.75' to each side of the equation.
4.75 + -14z + -4.75 + z2 = 0 + -4.75

Reorder the terms:
4.75 + -4.75 + -14z + z2 = 0 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + -14z + z2 = 0 + -4.75
-14z + z2 = 0 + -4.75

Combine like terms: 0 + -4.75 = -4.75
-14z + z2 = -4.75

The z term is -14z.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14z + 49 + z2 = -4.75 + 49

Reorder the terms:
49 + -14z + z2 = -4.75 + 49

Combine like terms: -4.75 + 49 = 44.25
49 + -14z + z2 = 44.25

Factor a perfect square on the left side:
(z + -7)(z + -7) = 44.25

Calculate the square root of the right side: 6.652067348

Break this problem into two subproblems by setting 
(z + -7) equal to 6.652067348 and -6.652067348.

Subproblem 1

z + -7 = 6.652067348 Simplifying z + -7 = 6.652067348 Reorder the terms: -7 + z = 6.652067348 Solving -7 + z = 6.652067348 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = 6.652067348 + 7 Combine like terms: -7 + 7 = 0 0 + z = 6.652067348 + 7 z = 6.652067348 + 7 Combine like terms: 6.652067348 + 7 = 13.652067348 z = 13.652067348 Simplifying z = 13.652067348

Subproblem 2

z + -7 = -6.652067348 Simplifying z + -7 = -6.652067348 Reorder the terms: -7 + z = -6.652067348 Solving -7 + z = -6.652067348 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = -6.652067348 + 7 Combine like terms: -7 + 7 = 0 0 + z = -6.652067348 + 7 z = -6.652067348 + 7 Combine like terms: -6.652067348 + 7 = 0.347932652 z = 0.347932652 Simplifying z = 0.347932652

Solution

The solution to the problem is based on the solutions from the subproblems. z = {13.652067348, 0.347932652}

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