3z+51=z^2-5z-49

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Solution for 3z+51=z^2-5z-49 equation:


Simplifying
3z + 51 = z2 + -5z + -49

Reorder the terms:
51 + 3z = z2 + -5z + -49

Reorder the terms:
51 + 3z = -49 + -5z + z2

Solving
51 + 3z = -49 + -5z + z2

Solving for variable 'z'.

Reorder the terms:
51 + 49 + 3z + 5z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2

Combine like terms: 51 + 49 = 100
100 + 3z + 5z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2

Combine like terms: 3z + 5z = 8z
100 + 8z + -1z2 = -49 + -5z + z2 + 49 + 5z + -1z2

Reorder the terms:
100 + 8z + -1z2 = -49 + 49 + -5z + 5z + z2 + -1z2

Combine like terms: -49 + 49 = 0
100 + 8z + -1z2 = 0 + -5z + 5z + z2 + -1z2
100 + 8z + -1z2 = -5z + 5z + z2 + -1z2

Combine like terms: -5z + 5z = 0
100 + 8z + -1z2 = 0 + z2 + -1z2
100 + 8z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
100 + 8z + -1z2 = 0

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

Divide each side by '-1'.
-100 + -8z + z2 = 0

Move the constant term to the right:

Add '100' to each side of the equation.
-100 + -8z + 100 + z2 = 0 + 100

Reorder the terms:
-100 + 100 + -8z + z2 = 0 + 100

Combine like terms: -100 + 100 = 0
0 + -8z + z2 = 0 + 100
-8z + z2 = 0 + 100

Combine like terms: 0 + 100 = 100
-8z + z2 = 100

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = 100 + 16

Reorder the terms:
16 + -8z + z2 = 100 + 16

Combine like terms: 100 + 16 = 116
16 + -8z + z2 = 116

Factor a perfect square on the left side:
(z + -4)(z + -4) = 116

Calculate the square root of the right side: 10.770329614

Break this problem into two subproblems by setting 
(z + -4) equal to 10.770329614 and -10.770329614.

Subproblem 1

z + -4 = 10.770329614 Simplifying z + -4 = 10.770329614 Reorder the terms: -4 + z = 10.770329614 Solving -4 + z = 10.770329614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = 10.770329614 + 4 Combine like terms: -4 + 4 = 0 0 + z = 10.770329614 + 4 z = 10.770329614 + 4 Combine like terms: 10.770329614 + 4 = 14.770329614 z = 14.770329614 Simplifying z = 14.770329614

Subproblem 2

z + -4 = -10.770329614 Simplifying z + -4 = -10.770329614 Reorder the terms: -4 + z = -10.770329614 Solving -4 + z = -10.770329614 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z = -10.770329614 + 4 Combine like terms: -4 + 4 = 0 0 + z = -10.770329614 + 4 z = -10.770329614 + 4 Combine like terms: -10.770329614 + 4 = -6.770329614 z = -6.770329614 Simplifying z = -6.770329614

Solution

The solution to the problem is based on the solutions from the subproblems. z = {14.770329614, -6.770329614}

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