3z^2+15z+10=0

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Solution for 3z^2+15z+10=0 equation:


Simplifying
3z2 + 15z + 10 = 0

Reorder the terms:
10 + 15z + 3z2 = 0

Solving
10 + 15z + 3z2 = 0

Solving for variable 'z'.

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

Divide each side by '3'.
3.333333333 + 5z + z2 = 0

Move the constant term to the right:

Add '-3.333333333' to each side of the equation.
3.333333333 + 5z + -3.333333333 + z2 = 0 + -3.333333333

Reorder the terms:
3.333333333 + -3.333333333 + 5z + z2 = 0 + -3.333333333

Combine like terms: 3.333333333 + -3.333333333 = 0.000000000
0.000000000 + 5z + z2 = 0 + -3.333333333
5z + z2 = 0 + -3.333333333

Combine like terms: 0 + -3.333333333 = -3.333333333
5z + z2 = -3.333333333

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 = -3.333333333 + 6.25

Reorder the terms:
6.25 + 5z + z2 = -3.333333333 + 6.25

Combine like terms: -3.333333333 + 6.25 = 2.916666667
6.25 + 5z + z2 = 2.916666667

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 2.916666667

Calculate the square root of the right side: 1.707825128

Break this problem into two subproblems by setting 
(z + 2.5) equal to 1.707825128 and -1.707825128.

Subproblem 1

z + 2.5 = 1.707825128 Simplifying z + 2.5 = 1.707825128 Reorder the terms: 2.5 + z = 1.707825128 Solving 2.5 + z = 1.707825128 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 = 1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 1.707825128 + -2.5 z = 1.707825128 + -2.5 Combine like terms: 1.707825128 + -2.5 = -0.792174872 z = -0.792174872 Simplifying z = -0.792174872

Subproblem 2

z + 2.5 = -1.707825128 Simplifying z + 2.5 = -1.707825128 Reorder the terms: 2.5 + z = -1.707825128 Solving 2.5 + z = -1.707825128 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 = -1.707825128 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -1.707825128 + -2.5 z = -1.707825128 + -2.5 Combine like terms: -1.707825128 + -2.5 = -4.207825128 z = -4.207825128 Simplifying z = -4.207825128

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.792174872, -4.207825128}

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