5z^2+19z=-4

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Solution for 5z^2+19z=-4 equation:


Simplifying
5z2 + 19z = -4

Reorder the terms:
19z + 5z2 = -4

Solving
19z + 5z2 = -4

Solving for variable 'z'.

Reorder the terms:
4 + 19z + 5z2 = -4 + 4

Combine like terms: -4 + 4 = 0
4 + 19z + 5z2 = 0

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

Divide each side by '5'.
0.8 + 3.8z + z2 = 0

Move the constant term to the right:

Add '-0.8' to each side of the equation.
0.8 + 3.8z + -0.8 + z2 = 0 + -0.8

Reorder the terms:
0.8 + -0.8 + 3.8z + z2 = 0 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + 3.8z + z2 = 0 + -0.8
3.8z + z2 = 0 + -0.8

Combine like terms: 0 + -0.8 = -0.8
3.8z + z2 = -0.8

The z term is 3.8z.  Take half its coefficient (1.9).
Square it (3.61) and add it to both sides.

Add '3.61' to each side of the equation.
3.8z + 3.61 + z2 = -0.8 + 3.61

Reorder the terms:
3.61 + 3.8z + z2 = -0.8 + 3.61

Combine like terms: -0.8 + 3.61 = 2.81
3.61 + 3.8z + z2 = 2.81

Factor a perfect square on the left side:
(z + 1.9)(z + 1.9) = 2.81

Calculate the square root of the right side: 1.676305461

Break this problem into two subproblems by setting 
(z + 1.9) equal to 1.676305461 and -1.676305461.

Subproblem 1

z + 1.9 = 1.676305461 Simplifying z + 1.9 = 1.676305461 Reorder the terms: 1.9 + z = 1.676305461 Solving 1.9 + z = 1.676305461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = 1.676305461 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = 1.676305461 + -1.9 z = 1.676305461 + -1.9 Combine like terms: 1.676305461 + -1.9 = -0.223694539 z = -0.223694539 Simplifying z = -0.223694539

Subproblem 2

z + 1.9 = -1.676305461 Simplifying z + 1.9 = -1.676305461 Reorder the terms: 1.9 + z = -1.676305461 Solving 1.9 + z = -1.676305461 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.9' to each side of the equation. 1.9 + -1.9 + z = -1.676305461 + -1.9 Combine like terms: 1.9 + -1.9 = 0.0 0.0 + z = -1.676305461 + -1.9 z = -1.676305461 + -1.9 Combine like terms: -1.676305461 + -1.9 = -3.576305461 z = -3.576305461 Simplifying z = -3.576305461

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.223694539, -3.576305461}

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