0=z^2+4z-20

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Solution for 0=z^2+4z-20 equation:


Simplifying
0 = z2 + 4z + -20

Reorder the terms:
0 = -20 + 4z + z2

Solving
0 = -20 + 4z + z2

Solving for variable 'z'.

Combine like terms: 0 + 20 = 20
20 + -4z + -1z2 = -20 + 4z + z2 + 20 + -4z + -1z2

Reorder the terms:
20 + -4z + -1z2 = -20 + 20 + 4z + -4z + z2 + -1z2

Combine like terms: -20 + 20 = 0
20 + -4z + -1z2 = 0 + 4z + -4z + z2 + -1z2
20 + -4z + -1z2 = 4z + -4z + z2 + -1z2

Combine like terms: 4z + -4z = 0
20 + -4z + -1z2 = 0 + z2 + -1z2
20 + -4z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
20 + -4z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-20 + 4z + z2 = 0

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 4z + 20 + z2 = 0 + 20

Reorder the terms:
-20 + 20 + 4z + z2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 4z + z2 = 0 + 20
4z + z2 = 0 + 20

Combine like terms: 0 + 20 = 20
4z + z2 = 20

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = 20 + 4

Reorder the terms:
4 + 4z + z2 = 20 + 4

Combine like terms: 20 + 4 = 24
4 + 4z + z2 = 24

Factor a perfect square on the left side:
(z + 2)(z + 2) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(z + 2) equal to 4.898979486 and -4.898979486.

Subproblem 1

z + 2 = 4.898979486 Simplifying z + 2 = 4.898979486 Reorder the terms: 2 + z = 4.898979486 Solving 2 + z = 4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 4.898979486 + -2 Combine like terms: 2 + -2 = 0 0 + z = 4.898979486 + -2 z = 4.898979486 + -2 Combine like terms: 4.898979486 + -2 = 2.898979486 z = 2.898979486 Simplifying z = 2.898979486

Subproblem 2

z + 2 = -4.898979486 Simplifying z + 2 = -4.898979486 Reorder the terms: 2 + z = -4.898979486 Solving 2 + z = -4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -4.898979486 + -2 Combine like terms: 2 + -2 = 0 0 + z = -4.898979486 + -2 z = -4.898979486 + -2 Combine like terms: -4.898979486 + -2 = -6.898979486 z = -6.898979486 Simplifying z = -6.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.898979486, -6.898979486}

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