z^2-6z-15=0

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Solution for z^2-6z-15=0 equation:


Simplifying
z2 + -6z + -15 = 0

Reorder the terms:
-15 + -6z + z2 = 0

Solving
-15 + -6z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + -6z + 15 + z2 = 0 + 15

Reorder the terms:
-15 + 15 + -6z + z2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + -6z + z2 = 0 + 15
-6z + z2 = 0 + 15

Combine like terms: 0 + 15 = 15
-6z + z2 = 15

The z term is -6z.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6z + 9 + z2 = 15 + 9

Reorder the terms:
9 + -6z + z2 = 15 + 9

Combine like terms: 15 + 9 = 24
9 + -6z + z2 = 24

Factor a perfect square on the left side:
(z + -3)(z + -3) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(z + -3) equal to 4.898979486 and -4.898979486.

Subproblem 1

z + -3 = 4.898979486 Simplifying z + -3 = 4.898979486 Reorder the terms: -3 + z = 4.898979486 Solving -3 + z = 4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + z = 4.898979486 + 3 z = 4.898979486 + 3 Combine like terms: 4.898979486 + 3 = 7.898979486 z = 7.898979486 Simplifying z = 7.898979486

Subproblem 2

z + -3 = -4.898979486 Simplifying z + -3 = -4.898979486 Reorder the terms: -3 + z = -4.898979486 Solving -3 + z = -4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -4.898979486 + 3 Combine like terms: -3 + 3 = 0 0 + z = -4.898979486 + 3 z = -4.898979486 + 3 Combine like terms: -4.898979486 + 3 = -1.898979486 z = -1.898979486 Simplifying z = -1.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.898979486, -1.898979486}

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