(z-5)(z)=(0.5z+7)2

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Solution for (z-5)(z)=(0.5z+7)2 equation:


Simplifying
(z + -5)(z) = (0.5z + 7) * 2

Reorder the terms:
(-5 + z)(z) = (0.5z + 7) * 2

Reorder the terms for easier multiplication:
z(-5 + z) = (0.5z + 7) * 2
(-5 * z + z * z) = (0.5z + 7) * 2
(-5z + z2) = (0.5z + 7) * 2

Reorder the terms:
-5z + z2 = (7 + 0.5z) * 2

Reorder the terms for easier multiplication:
-5z + z2 = 2(7 + 0.5z)
-5z + z2 = (7 * 2 + 0.5z * 2)
-5z + z2 = (14 + 1z)

Solving
-5z + z2 = 14 + 1z

Solving for variable 'z'.

Reorder the terms:
-14 + -5z + -1z + z2 = 14 + 1z + -14 + -1z

Combine like terms: -5z + -1z = -6z
-14 + -6z + z2 = 14 + 1z + -14 + -1z

Reorder the terms:
-14 + -6z + z2 = 14 + -14 + 1z + -1z

Combine like terms: 14 + -14 = 0
-14 + -6z + z2 = 0 + 1z + -1z
-14 + -6z + z2 = 1z + -1z

Combine like terms: 1z + -1z = 0
-14 + -6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + -6z + 14 + z2 = 0 + 14

Reorder the terms:
-14 + 14 + -6z + z2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + -6z + z2 = 0 + 14
-6z + z2 = 0 + 14

Combine like terms: 0 + 14 = 14
-6z + z2 = 14

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 = 14 + 9

Reorder the terms:
9 + -6z + z2 = 14 + 9

Combine like terms: 14 + 9 = 23
9 + -6z + z2 = 23

Factor a perfect square on the left side:
(z + -3)(z + -3) = 23

Calculate the square root of the right side: 4.795831523

Break this problem into two subproblems by setting 
(z + -3) equal to 4.795831523 and -4.795831523.

Subproblem 1

z + -3 = 4.795831523 Simplifying z + -3 = 4.795831523 Reorder the terms: -3 + z = 4.795831523 Solving -3 + z = 4.795831523 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 = 4.795831523 + 3 Combine like terms: -3 + 3 = 0 0 + z = 4.795831523 + 3 z = 4.795831523 + 3 Combine like terms: 4.795831523 + 3 = 7.795831523 z = 7.795831523 Simplifying z = 7.795831523

Subproblem 2

z + -3 = -4.795831523 Simplifying z + -3 = -4.795831523 Reorder the terms: -3 + z = -4.795831523 Solving -3 + z = -4.795831523 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 = -4.795831523 + 3 Combine like terms: -3 + 3 = 0 0 + z = -4.795831523 + 3 z = -4.795831523 + 3 Combine like terms: -4.795831523 + 3 = -1.795831523 z = -1.795831523 Simplifying z = -1.795831523

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.795831523, -1.795831523}

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