24z^2+49z-65=0

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Solution for 24z^2+49z-65=0 equation:


Simplifying
24z2 + 49z + -65 = 0

Reorder the terms:
-65 + 49z + 24z2 = 0

Solving
-65 + 49z + 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'.
-2.708333333 + 2.041666667z + z2 = 0

Move the constant term to the right:

Add '2.708333333' to each side of the equation.
-2.708333333 + 2.041666667z + 2.708333333 + z2 = 0 + 2.708333333

Reorder the terms:
-2.708333333 + 2.708333333 + 2.041666667z + z2 = 0 + 2.708333333

Combine like terms: -2.708333333 + 2.708333333 = 0.000000000
0.000000000 + 2.041666667z + z2 = 0 + 2.708333333
2.041666667z + z2 = 0 + 2.708333333

Combine like terms: 0 + 2.708333333 = 2.708333333
2.041666667z + z2 = 2.708333333

The z term is 2.041666667z.  Take half its coefficient (1.020833334).
Square it (1.042100696) and add it to both sides.

Add '1.042100696' to each side of the equation.
2.041666667z + 1.042100696 + z2 = 2.708333333 + 1.042100696

Reorder the terms:
1.042100696 + 2.041666667z + z2 = 2.708333333 + 1.042100696

Combine like terms: 2.708333333 + 1.042100696 = 3.750434029
1.042100696 + 2.041666667z + z2 = 3.750434029

Factor a perfect square on the left side:
(z + 1.020833334)(z + 1.020833334) = 3.750434029

Calculate the square root of the right side: 1.936603736

Break this problem into two subproblems by setting 
(z + 1.020833334) equal to 1.936603736 and -1.936603736.

Subproblem 1

z + 1.020833334 = 1.936603736 Simplifying z + 1.020833334 = 1.936603736 Reorder the terms: 1.020833334 + z = 1.936603736 Solving 1.020833334 + z = 1.936603736 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + z = 1.936603736 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + z = 1.936603736 + -1.020833334 z = 1.936603736 + -1.020833334 Combine like terms: 1.936603736 + -1.020833334 = 0.915770402 z = 0.915770402 Simplifying z = 0.915770402

Subproblem 2

z + 1.020833334 = -1.936603736 Simplifying z + 1.020833334 = -1.936603736 Reorder the terms: 1.020833334 + z = -1.936603736 Solving 1.020833334 + z = -1.936603736 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.020833334' to each side of the equation. 1.020833334 + -1.020833334 + z = -1.936603736 + -1.020833334 Combine like terms: 1.020833334 + -1.020833334 = 0.000000000 0.000000000 + z = -1.936603736 + -1.020833334 z = -1.936603736 + -1.020833334 Combine like terms: -1.936603736 + -1.020833334 = -2.95743707 z = -2.95743707 Simplifying z = -2.95743707

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.915770402, -2.95743707}

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