2z^2+9z-15=0

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Solution for 2z^2+9z-15=0 equation:


Simplifying
2z2 + 9z + -15 = 0

Reorder the terms:
-15 + 9z + 2z2 = 0

Solving
-15 + 9z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
-7.5 + 4.5z + z2 = 0

Move the constant term to the right:

Add '7.5' to each side of the equation.
-7.5 + 4.5z + 7.5 + z2 = 0 + 7.5

Reorder the terms:
-7.5 + 7.5 + 4.5z + z2 = 0 + 7.5

Combine like terms: -7.5 + 7.5 = 0.0
0.0 + 4.5z + z2 = 0 + 7.5
4.5z + z2 = 0 + 7.5

Combine like terms: 0 + 7.5 = 7.5
4.5z + z2 = 7.5

The z term is 4.5z.  Take half its coefficient (2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
4.5z + 5.0625 + z2 = 7.5 + 5.0625

Reorder the terms:
5.0625 + 4.5z + z2 = 7.5 + 5.0625

Combine like terms: 7.5 + 5.0625 = 12.5625
5.0625 + 4.5z + z2 = 12.5625

Factor a perfect square on the left side:
(z + 2.25)(z + 2.25) = 12.5625

Calculate the square root of the right side: 3.54436172

Break this problem into two subproblems by setting 
(z + 2.25) equal to 3.54436172 and -3.54436172.

Subproblem 1

z + 2.25 = 3.54436172 Simplifying z + 2.25 = 3.54436172 Reorder the terms: 2.25 + z = 3.54436172 Solving 2.25 + z = 3.54436172 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = 3.54436172 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = 3.54436172 + -2.25 z = 3.54436172 + -2.25 Combine like terms: 3.54436172 + -2.25 = 1.29436172 z = 1.29436172 Simplifying z = 1.29436172

Subproblem 2

z + 2.25 = -3.54436172 Simplifying z + 2.25 = -3.54436172 Reorder the terms: 2.25 + z = -3.54436172 Solving 2.25 + z = -3.54436172 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + z = -3.54436172 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + z = -3.54436172 + -2.25 z = -3.54436172 + -2.25 Combine like terms: -3.54436172 + -2.25 = -5.79436172 z = -5.79436172 Simplifying z = -5.79436172

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.29436172, -5.79436172}

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