z^2+7z-40=0

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Solution for z^2+7z-40=0 equation:


Simplifying
z2 + 7z + -40 = 0

Reorder the terms:
-40 + 7z + z2 = 0

Solving
-40 + 7z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 7z + 40 + z2 = 0 + 40

Reorder the terms:
-40 + 40 + 7z + z2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 7z + z2 = 0 + 40
7z + z2 = 0 + 40

Combine like terms: 0 + 40 = 40
7z + z2 = 40

The z term is 7z.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7z + 12.25 + z2 = 40 + 12.25

Reorder the terms:
12.25 + 7z + z2 = 40 + 12.25

Combine like terms: 40 + 12.25 = 52.25
12.25 + 7z + z2 = 52.25

Factor a perfect square on the left side:
(z + 3.5)(z + 3.5) = 52.25

Calculate the square root of the right side: 7.228416147

Break this problem into two subproblems by setting 
(z + 3.5) equal to 7.228416147 and -7.228416147.

Subproblem 1

z + 3.5 = 7.228416147 Simplifying z + 3.5 = 7.228416147 Reorder the terms: 3.5 + z = 7.228416147 Solving 3.5 + z = 7.228416147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = 7.228416147 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = 7.228416147 + -3.5 z = 7.228416147 + -3.5 Combine like terms: 7.228416147 + -3.5 = 3.728416147 z = 3.728416147 Simplifying z = 3.728416147

Subproblem 2

z + 3.5 = -7.228416147 Simplifying z + 3.5 = -7.228416147 Reorder the terms: 3.5 + z = -7.228416147 Solving 3.5 + z = -7.228416147 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + z = -7.228416147 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + z = -7.228416147 + -3.5 z = -7.228416147 + -3.5 Combine like terms: -7.228416147 + -3.5 = -10.728416147 z = -10.728416147 Simplifying z = -10.728416147

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.728416147, -10.728416147}

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