3z^2+z-2=2z+7

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Solution for 3z^2+z-2=2z+7 equation:


Simplifying
3z2 + z + -2 = 2z + 7

Reorder the terms:
-2 + z + 3z2 = 2z + 7

Reorder the terms:
-2 + z + 3z2 = 7 + 2z

Solving
-2 + z + 3z2 = 7 + 2z

Solving for variable 'z'.

Reorder the terms:
-2 + -7 + z + -2z + 3z2 = 7 + 2z + -7 + -2z

Combine like terms: -2 + -7 = -9
-9 + z + -2z + 3z2 = 7 + 2z + -7 + -2z

Combine like terms: z + -2z = -1z
-9 + -1z + 3z2 = 7 + 2z + -7 + -2z

Reorder the terms:
-9 + -1z + 3z2 = 7 + -7 + 2z + -2z

Combine like terms: 7 + -7 = 0
-9 + -1z + 3z2 = 0 + 2z + -2z
-9 + -1z + 3z2 = 2z + -2z

Combine like terms: 2z + -2z = 0
-9 + -1z + 3z2 = 0

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

Divide each side by '3'.
-3 + -0.3333333333z + z2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.3333333333z + 3 + z2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.3333333333z + z2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.3333333333z + z2 = 0 + 3
-0.3333333333z + z2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.3333333333z + z2 = 3

The z term is -0.3333333333z.  Take half its coefficient (-0.1666666667).
Square it (0.02777777779) and add it to both sides.

Add '0.02777777779' to each side of the equation.
-0.3333333333z + 0.02777777779 + z2 = 3 + 0.02777777779

Reorder the terms:
0.02777777779 + -0.3333333333z + z2 = 3 + 0.02777777779

Combine like terms: 3 + 0.02777777779 = 3.02777777779
0.02777777779 + -0.3333333333z + z2 = 3.02777777779

Factor a perfect square on the left side:
(z + -0.1666666667)(z + -0.1666666667) = 3.02777777779

Calculate the square root of the right side: 1.740051085

Break this problem into two subproblems by setting 
(z + -0.1666666667) equal to 1.740051085 and -1.740051085.

Subproblem 1

z + -0.1666666667 = 1.740051085 Simplifying z + -0.1666666667 = 1.740051085 Reorder the terms: -0.1666666667 + z = 1.740051085 Solving -0.1666666667 + z = 1.740051085 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + z = 1.740051085 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + z = 1.740051085 + 0.1666666667 z = 1.740051085 + 0.1666666667 Combine like terms: 1.740051085 + 0.1666666667 = 1.9067177517 z = 1.9067177517 Simplifying z = 1.9067177517

Subproblem 2

z + -0.1666666667 = -1.740051085 Simplifying z + -0.1666666667 = -1.740051085 Reorder the terms: -0.1666666667 + z = -1.740051085 Solving -0.1666666667 + z = -1.740051085 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.1666666667' to each side of the equation. -0.1666666667 + 0.1666666667 + z = -1.740051085 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + z = -1.740051085 + 0.1666666667 z = -1.740051085 + 0.1666666667 Combine like terms: -1.740051085 + 0.1666666667 = -1.5733844183 z = -1.5733844183 Simplifying z = -1.5733844183

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.9067177517, -1.5733844183}

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