7z-10=z^2-7z+2

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Solution for 7z-10=z^2-7z+2 equation:


Simplifying
7z + -10 = z2 + -7z + 2

Reorder the terms:
-10 + 7z = z2 + -7z + 2

Reorder the terms:
-10 + 7z = 2 + -7z + z2

Solving
-10 + 7z = 2 + -7z + z2

Solving for variable 'z'.

Reorder the terms:
-10 + -2 + 7z + 7z + -1z2 = 2 + -7z + z2 + -2 + 7z + -1z2

Combine like terms: -10 + -2 = -12
-12 + 7z + 7z + -1z2 = 2 + -7z + z2 + -2 + 7z + -1z2

Combine like terms: 7z + 7z = 14z
-12 + 14z + -1z2 = 2 + -7z + z2 + -2 + 7z + -1z2

Reorder the terms:
-12 + 14z + -1z2 = 2 + -2 + -7z + 7z + z2 + -1z2

Combine like terms: 2 + -2 = 0
-12 + 14z + -1z2 = 0 + -7z + 7z + z2 + -1z2
-12 + 14z + -1z2 = -7z + 7z + z2 + -1z2

Combine like terms: -7z + 7z = 0
-12 + 14z + -1z2 = 0 + z2 + -1z2
-12 + 14z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
-12 + 14z + -1z2 = 0

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

Divide each side by '-1'.
12 + -14z + z2 = 0

Move the constant term to the right:

Add '-12' to each side of the equation.
12 + -14z + -12 + z2 = 0 + -12

Reorder the terms:
12 + -12 + -14z + z2 = 0 + -12

Combine like terms: 12 + -12 = 0
0 + -14z + z2 = 0 + -12
-14z + z2 = 0 + -12

Combine like terms: 0 + -12 = -12
-14z + z2 = -12

The z term is -14z.  Take half its coefficient (-7).
Square it (49) and add it to both sides.

Add '49' to each side of the equation.
-14z + 49 + z2 = -12 + 49

Reorder the terms:
49 + -14z + z2 = -12 + 49

Combine like terms: -12 + 49 = 37
49 + -14z + z2 = 37

Factor a perfect square on the left side:
(z + -7)(z + -7) = 37

Calculate the square root of the right side: 6.08276253

Break this problem into two subproblems by setting 
(z + -7) equal to 6.08276253 and -6.08276253.

Subproblem 1

z + -7 = 6.08276253 Simplifying z + -7 = 6.08276253 Reorder the terms: -7 + z = 6.08276253 Solving -7 + z = 6.08276253 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = 6.08276253 + 7 Combine like terms: -7 + 7 = 0 0 + z = 6.08276253 + 7 z = 6.08276253 + 7 Combine like terms: 6.08276253 + 7 = 13.08276253 z = 13.08276253 Simplifying z = 13.08276253

Subproblem 2

z + -7 = -6.08276253 Simplifying z + -7 = -6.08276253 Reorder the terms: -7 + z = -6.08276253 Solving -7 + z = -6.08276253 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + z = -6.08276253 + 7 Combine like terms: -7 + 7 = 0 0 + z = -6.08276253 + 7 z = -6.08276253 + 7 Combine like terms: -6.08276253 + 7 = 0.91723747 z = 0.91723747 Simplifying z = 0.91723747

Solution

The solution to the problem is based on the solutions from the subproblems. z = {13.08276253, 0.91723747}

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