24z^2+55z+24=0

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Solution for 24z^2+55z+24=0 equation:


Simplifying
24z2 + 55z + 24 = 0

Reorder the terms:
24 + 55z + 24z2 = 0

Solving
24 + 55z + 24z2 = 0

Solving for variable 'z'.

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

Divide each side by '24'.
1 + 2.291666667z + z2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.291666667z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + 2.291666667z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.291666667z + z2 = 0 + -1
2.291666667z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.291666667z + z2 = -1

The z term is 2.291666667z.  Take half its coefficient (1.145833334).
Square it (1.312934029) and add it to both sides.

Add '1.312934029' to each side of the equation.
2.291666667z + 1.312934029 + z2 = -1 + 1.312934029

Reorder the terms:
1.312934029 + 2.291666667z + z2 = -1 + 1.312934029

Combine like terms: -1 + 1.312934029 = 0.312934029
1.312934029 + 2.291666667z + z2 = 0.312934029

Factor a perfect square on the left side:
(z + 1.145833334)(z + 1.145833334) = 0.312934029

Calculate the square root of the right side: 0.559405067

Break this problem into two subproblems by setting 
(z + 1.145833334) equal to 0.559405067 and -0.559405067.

Subproblem 1

z + 1.145833334 = 0.559405067 Simplifying z + 1.145833334 = 0.559405067 Reorder the terms: 1.145833334 + z = 0.559405067 Solving 1.145833334 + z = 0.559405067 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.145833334' to each side of the equation. 1.145833334 + -1.145833334 + z = 0.559405067 + -1.145833334 Combine like terms: 1.145833334 + -1.145833334 = 0.000000000 0.000000000 + z = 0.559405067 + -1.145833334 z = 0.559405067 + -1.145833334 Combine like terms: 0.559405067 + -1.145833334 = -0.586428267 z = -0.586428267 Simplifying z = -0.586428267

Subproblem 2

z + 1.145833334 = -0.559405067 Simplifying z + 1.145833334 = -0.559405067 Reorder the terms: 1.145833334 + z = -0.559405067 Solving 1.145833334 + z = -0.559405067 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.145833334' to each side of the equation. 1.145833334 + -1.145833334 + z = -0.559405067 + -1.145833334 Combine like terms: 1.145833334 + -1.145833334 = 0.000000000 0.000000000 + z = -0.559405067 + -1.145833334 z = -0.559405067 + -1.145833334 Combine like terms: -0.559405067 + -1.145833334 = -1.705238401 z = -1.705238401 Simplifying z = -1.705238401

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-0.586428267, -1.705238401}

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