14z+8z^2=27

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Solution for 14z+8z^2=27 equation:


Simplifying
14z + 8z2 = 27

Solving
14z + 8z2 = 27

Solving for variable 'z'.

Reorder the terms:
-27 + 14z + 8z2 = 27 + -27

Combine like terms: 27 + -27 = 0
-27 + 14z + 8z2 = 0

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

Divide each side by '8'.
-3.375 + 1.75z + z2 = 0

Move the constant term to the right:

Add '3.375' to each side of the equation.
-3.375 + 1.75z + 3.375 + z2 = 0 + 3.375

Reorder the terms:
-3.375 + 3.375 + 1.75z + z2 = 0 + 3.375

Combine like terms: -3.375 + 3.375 = 0.000
0.000 + 1.75z + z2 = 0 + 3.375
1.75z + z2 = 0 + 3.375

Combine like terms: 0 + 3.375 = 3.375
1.75z + z2 = 3.375

The z term is 1.75z.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75z + 0.765625 + z2 = 3.375 + 0.765625

Reorder the terms:
0.765625 + 1.75z + z2 = 3.375 + 0.765625

Combine like terms: 3.375 + 0.765625 = 4.140625
0.765625 + 1.75z + z2 = 4.140625

Factor a perfect square on the left side:
(z + 0.875)(z + 0.875) = 4.140625

Calculate the square root of the right side: 2.034852575

Break this problem into two subproblems by setting 
(z + 0.875) equal to 2.034852575 and -2.034852575.

Subproblem 1

z + 0.875 = 2.034852575 Simplifying z + 0.875 = 2.034852575 Reorder the terms: 0.875 + z = 2.034852575 Solving 0.875 + z = 2.034852575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + z = 2.034852575 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + z = 2.034852575 + -0.875 z = 2.034852575 + -0.875 Combine like terms: 2.034852575 + -0.875 = 1.159852575 z = 1.159852575 Simplifying z = 1.159852575

Subproblem 2

z + 0.875 = -2.034852575 Simplifying z + 0.875 = -2.034852575 Reorder the terms: 0.875 + z = -2.034852575 Solving 0.875 + z = -2.034852575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + z = -2.034852575 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + z = -2.034852575 + -0.875 z = -2.034852575 + -0.875 Combine like terms: -2.034852575 + -0.875 = -2.909852575 z = -2.909852575 Simplifying z = -2.909852575

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.159852575, -2.909852575}

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