4z^2+13=16z

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Solution for 4z^2+13=16z equation:


Simplifying
4z2 + 13 = 16z

Reorder the terms:
13 + 4z2 = 16z

Solving
13 + 4z2 = 16z

Solving for variable 'z'.

Reorder the terms:
13 + -16z + 4z2 = 16z + -16z

Combine like terms: 16z + -16z = 0
13 + -16z + 4z2 = 0

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

Divide each side by '4'.
3.25 + -4z + z2 = 0

Move the constant term to the right:

Add '-3.25' to each side of the equation.
3.25 + -4z + -3.25 + z2 = 0 + -3.25

Reorder the terms:
3.25 + -3.25 + -4z + z2 = 0 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + -4z + z2 = 0 + -3.25
-4z + z2 = 0 + -3.25

Combine like terms: 0 + -3.25 = -3.25
-4z + z2 = -3.25

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = -3.25 + 4

Reorder the terms:
4 + -4z + z2 = -3.25 + 4

Combine like terms: -3.25 + 4 = 0.75
4 + -4z + z2 = 0.75

Factor a perfect square on the left side:
(z + -2)(z + -2) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(z + -2) equal to 0.866025404 and -0.866025404.

Subproblem 1

z + -2 = 0.866025404 Simplifying z + -2 = 0.866025404 Reorder the terms: -2 + z = 0.866025404 Solving -2 + z = 0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = 0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + z = 0.866025404 + 2 z = 0.866025404 + 2 Combine like terms: 0.866025404 + 2 = 2.866025404 z = 2.866025404 Simplifying z = 2.866025404

Subproblem 2

z + -2 = -0.866025404 Simplifying z + -2 = -0.866025404 Reorder the terms: -2 + z = -0.866025404 Solving -2 + z = -0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + z = -0.866025404 + 2 Combine like terms: -2 + 2 = 0 0 + z = -0.866025404 + 2 z = -0.866025404 + 2 Combine like terms: -0.866025404 + 2 = 1.133974596 z = 1.133974596 Simplifying z = 1.133974596

Solution

The solution to the problem is based on the solutions from the subproblems. z = {2.866025404, 1.133974596}

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