2z^2-19z+3=0

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Solution for 2z^2-19z+3=0 equation:


Simplifying
2z2 + -19z + 3 = 0

Reorder the terms:
3 + -19z + 2z2 = 0

Solving
3 + -19z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
1.5 + -9.5z + z2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -9.5z + -1.5 + z2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -9.5z + z2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -9.5z + z2 = 0 + -1.5
-9.5z + z2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-9.5z + z2 = -1.5

The z term is -9.5z.  Take half its coefficient (-4.75).
Square it (22.5625) and add it to both sides.

Add '22.5625' to each side of the equation.
-9.5z + 22.5625 + z2 = -1.5 + 22.5625

Reorder the terms:
22.5625 + -9.5z + z2 = -1.5 + 22.5625

Combine like terms: -1.5 + 22.5625 = 21.0625
22.5625 + -9.5z + z2 = 21.0625

Factor a perfect square on the left side:
(z + -4.75)(z + -4.75) = 21.0625

Calculate the square root of the right side: 4.589389938

Break this problem into two subproblems by setting 
(z + -4.75) equal to 4.589389938 and -4.589389938.

Subproblem 1

z + -4.75 = 4.589389938 Simplifying z + -4.75 = 4.589389938 Reorder the terms: -4.75 + z = 4.589389938 Solving -4.75 + z = 4.589389938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.75' to each side of the equation. -4.75 + 4.75 + z = 4.589389938 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + z = 4.589389938 + 4.75 z = 4.589389938 + 4.75 Combine like terms: 4.589389938 + 4.75 = 9.339389938 z = 9.339389938 Simplifying z = 9.339389938

Subproblem 2

z + -4.75 = -4.589389938 Simplifying z + -4.75 = -4.589389938 Reorder the terms: -4.75 + z = -4.589389938 Solving -4.75 + z = -4.589389938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.75' to each side of the equation. -4.75 + 4.75 + z = -4.589389938 + 4.75 Combine like terms: -4.75 + 4.75 = 0.00 0.00 + z = -4.589389938 + 4.75 z = -4.589389938 + 4.75 Combine like terms: -4.589389938 + 4.75 = 0.160610062 z = 0.160610062 Simplifying z = 0.160610062

Solution

The solution to the problem is based on the solutions from the subproblems. z = {9.339389938, 0.160610062}

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