z^2-24z=47

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Solution for z^2-24z=47 equation:


Simplifying
z2 + -24z = 47

Reorder the terms:
-24z + z2 = 47

Solving
-24z + z2 = 47

Solving for variable 'z'.

Reorder the terms:
-47 + -24z + z2 = 47 + -47

Combine like terms: 47 + -47 = 0
-47 + -24z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '47' to each side of the equation.
-47 + -24z + 47 + z2 = 0 + 47

Reorder the terms:
-47 + 47 + -24z + z2 = 0 + 47

Combine like terms: -47 + 47 = 0
0 + -24z + z2 = 0 + 47
-24z + z2 = 0 + 47

Combine like terms: 0 + 47 = 47
-24z + z2 = 47

The z term is -24z.  Take half its coefficient (-12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
-24z + 144 + z2 = 47 + 144

Reorder the terms:
144 + -24z + z2 = 47 + 144

Combine like terms: 47 + 144 = 191
144 + -24z + z2 = 191

Factor a perfect square on the left side:
(z + -12)(z + -12) = 191

Calculate the square root of the right side: 13.820274961

Break this problem into two subproblems by setting 
(z + -12) equal to 13.820274961 and -13.820274961.

Subproblem 1

z + -12 = 13.820274961 Simplifying z + -12 = 13.820274961 Reorder the terms: -12 + z = 13.820274961 Solving -12 + z = 13.820274961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = 13.820274961 + 12 Combine like terms: -12 + 12 = 0 0 + z = 13.820274961 + 12 z = 13.820274961 + 12 Combine like terms: 13.820274961 + 12 = 25.820274961 z = 25.820274961 Simplifying z = 25.820274961

Subproblem 2

z + -12 = -13.820274961 Simplifying z + -12 = -13.820274961 Reorder the terms: -12 + z = -13.820274961 Solving -12 + z = -13.820274961 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = -13.820274961 + 12 Combine like terms: -12 + 12 = 0 0 + z = -13.820274961 + 12 z = -13.820274961 + 12 Combine like terms: -13.820274961 + 12 = -1.820274961 z = -1.820274961 Simplifying z = -1.820274961

Solution

The solution to the problem is based on the solutions from the subproblems. z = {25.820274961, -1.820274961}

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