4z^2-15z+4=0

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


Simplifying
4z2 + -15z + 4 = 0

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

Solving
4 + -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 + -3.75z + z2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -3.75z + -1 + z2 = 0 + -1

Reorder the terms:
1 + -1 + -3.75z + z2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -3.75z + z2 = 0 + -1
-3.75z + z2 = 0 + -1

Combine like terms: 0 + -1 = -1
-3.75z + z2 = -1

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 + 3.515625

Reorder the terms:
3.515625 + -3.75z + z2 = -1 + 3.515625

Combine like terms: -1 + 3.515625 = 2.515625
3.515625 + -3.75z + z2 = 2.515625

Factor a perfect square on the left side:
(z + -1.875)(z + -1.875) = 2.515625

Calculate the square root of the right side: 1.586072193

Break this problem into two subproblems by setting 
(z + -1.875) equal to 1.586072193 and -1.586072193.

Subproblem 1

z + -1.875 = 1.586072193 Simplifying z + -1.875 = 1.586072193 Reorder the terms: -1.875 + z = 1.586072193 Solving -1.875 + z = 1.586072193 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 = 1.586072193 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 1.586072193 + 1.875 z = 1.586072193 + 1.875 Combine like terms: 1.586072193 + 1.875 = 3.461072193 z = 3.461072193 Simplifying z = 3.461072193

Subproblem 2

z + -1.875 = -1.586072193 Simplifying z + -1.875 = -1.586072193 Reorder the terms: -1.875 + z = -1.586072193 Solving -1.875 + z = -1.586072193 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 = -1.586072193 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -1.586072193 + 1.875 z = -1.586072193 + 1.875 Combine like terms: -1.586072193 + 1.875 = 0.288927807 z = 0.288927807 Simplifying z = 0.288927807

Solution

The solution to the problem is based on the solutions from the subproblems. z = {3.461072193, 0.288927807}

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