z^2+5z+3=0

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Solution for z^2+5z+3=0 equation:


Simplifying
z2 + 5z + 3 = 0

Reorder the terms:
3 + 5z + z2 = 0

Solving
3 + 5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + 5z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5z + z2 = 0 + -3
5z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
5z + z2 = -3

The z term is 5z.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5z + 6.25 + z2 = -3 + 6.25

Reorder the terms:
6.25 + 5z + z2 = -3 + 6.25

Combine like terms: -3 + 6.25 = 3.25
6.25 + 5z + z2 = 3.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(z + 2.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

z + 2.5 = 1.802775638 Simplifying z + 2.5 = 1.802775638 Reorder the terms: 2.5 + z = 1.802775638 Solving 2.5 + z = 1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 1.802775638 + -2.5 z = 1.802775638 + -2.5 Combine like terms: 1.802775638 + -2.5 = -0.697224362 z = -0.697224362 Simplifying z = -0.697224362

Subproblem 2

z + 2.5 = -1.802775638 Simplifying z + 2.5 = -1.802775638 Reorder the terms: 2.5 + z = -1.802775638 Solving 2.5 + z = -1.802775638 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -1.802775638 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -1.802775638 + -2.5 z = -1.802775638 + -2.5 Combine like terms: -1.802775638 + -2.5 = -4.302775638 z = -4.302775638 Simplifying z = -4.302775638

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.697224362, -4.302775638}

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