z^2+5z-10=0

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Solution for z^2+5z-10=0 equation:


Simplifying
z2 + 5z + -10 = 0

Reorder the terms:
-10 + 5z + z2 = 0

Solving
-10 + 5z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + 5z + 10 + z2 = 0 + 10

Reorder the terms:
-10 + 10 + 5z + z2 = 0 + 10

Combine like terms: -10 + 10 = 0
0 + 5z + z2 = 0 + 10
5z + z2 = 0 + 10

Combine like terms: 0 + 10 = 10
5z + z2 = 10

The z term is 5z.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5z + 6.25 + z2 = 10 + 6.25

Reorder the terms:
6.25 + 5z + z2 = 10 + 6.25

Combine like terms: 10 + 6.25 = 16.25
6.25 + 5z + z2 = 16.25

Factor a perfect square on the left side:
(z + 2.5)(z + 2.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(z + 2.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

z + 2.5 = 4.031128874 Simplifying z + 2.5 = 4.031128874 Reorder the terms: 2.5 + z = 4.031128874 Solving 2.5 + z = 4.031128874 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = 4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = 4.031128874 + -2.5 z = 4.031128874 + -2.5 Combine like terms: 4.031128874 + -2.5 = 1.531128874 z = 1.531128874 Simplifying z = 1.531128874

Subproblem 2

z + 2.5 = -4.031128874 Simplifying z + 2.5 = -4.031128874 Reorder the terms: 2.5 + z = -4.031128874 Solving 2.5 + z = -4.031128874 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + z = -4.031128874 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + z = -4.031128874 + -2.5 z = -4.031128874 + -2.5 Combine like terms: -4.031128874 + -2.5 = -6.531128874 z = -6.531128874 Simplifying z = -6.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.531128874, -6.531128874}

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