4z^2+9z+3=0

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Solution for 4z^2+9z+3=0 equation:


Simplifying
4z2 + 9z + 3 = 0

Reorder the terms:
3 + 9z + 4z2 = 0

Solving
3 + 9z + 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'.
0.75 + 2.25z + z2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.25z + -0.75 + z2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.25z + z2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.25z + z2 = 0 + -0.75
2.25z + z2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.25z + z2 = -0.75

The z term is 2.25z.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25z + 1.265625 + z2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25z + z2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + 2.25z + z2 = 0.515625

Factor a perfect square on the left side:
(z + 1.125)(z + 1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(z + 1.125) equal to 0.718070331 and -0.718070331.

Subproblem 1

z + 1.125 = 0.718070331 Simplifying z + 1.125 = 0.718070331 Reorder the terms: 1.125 + z = 0.718070331 Solving 1.125 + z = 0.718070331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = 0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = 0.718070331 + -1.125 z = 0.718070331 + -1.125 Combine like terms: 0.718070331 + -1.125 = -0.406929669 z = -0.406929669 Simplifying z = -0.406929669

Subproblem 2

z + 1.125 = -0.718070331 Simplifying z + 1.125 = -0.718070331 Reorder the terms: 1.125 + z = -0.718070331 Solving 1.125 + z = -0.718070331 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + z = -0.718070331 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + z = -0.718070331 + -1.125 z = -0.718070331 + -1.125 Combine like terms: -0.718070331 + -1.125 = -1.843070331 z = -1.843070331 Simplifying z = -1.843070331

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.406929669, -1.843070331}

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