4z^2-15z-6=0

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Solution for 4z^2-15z-6=0 equation:


Simplifying
4z2 + -15z + -6 = 0

Reorder the terms:
-6 + -15z + 4z2 = 0

Solving
-6 + -15z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
-1.5 + -3.75z + z2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -3.75z + 1.5 + z2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -3.75z + z2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -3.75z + z2 = 0 + 1.5
-3.75z + z2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-3.75z + z2 = 1.5

The z term is -3.75z.  Take half its coefficient (-1.875).
Square it (3.515625) and add it to both sides.

Add '3.515625' to each side of the equation.
-3.75z + 3.515625 + z2 = 1.5 + 3.515625

Reorder the terms:
3.515625 + -3.75z + z2 = 1.5 + 3.515625

Combine like terms: 1.5 + 3.515625 = 5.015625
3.515625 + -3.75z + z2 = 5.015625

Factor a perfect square on the left side:
(z + -1.875)(z + -1.875) = 5.015625

Calculate the square root of the right side: 2.239559108

Break this problem into two subproblems by setting 
(z + -1.875) equal to 2.239559108 and -2.239559108.

Subproblem 1

z + -1.875 = 2.239559108 Simplifying z + -1.875 = 2.239559108 Reorder the terms: -1.875 + z = 2.239559108 Solving -1.875 + z = 2.239559108 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = 2.239559108 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 2.239559108 + 1.875 z = 2.239559108 + 1.875 Combine like terms: 2.239559108 + 1.875 = 4.114559108 z = 4.114559108 Simplifying z = 4.114559108

Subproblem 2

z + -1.875 = -2.239559108 Simplifying z + -1.875 = -2.239559108 Reorder the terms: -1.875 + z = -2.239559108 Solving -1.875 + z = -2.239559108 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = -2.239559108 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -2.239559108 + 1.875 z = -2.239559108 + 1.875 Combine like terms: -2.239559108 + 1.875 = -0.364559108 z = -0.364559108 Simplifying z = -0.364559108

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.114559108, -0.364559108}

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