4z^2-40z+76=0

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Solution for 4z^2-40z+76=0 equation:


Simplifying
4z2 + -40z + 76 = 0

Reorder the terms:
76 + -40z + 4z2 = 0

Solving
76 + -40z + 4z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '4'.
4(19 + -10z + z2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(19 + -10z + z2)' equal to zero and attempt to solve:

Simplifying
19 + -10z + z2 = 0

Solving
19 + -10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + -10z + -19 + z2 = 0 + -19

Reorder the terms:
19 + -19 + -10z + z2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + -10z + z2 = 0 + -19
-10z + z2 = 0 + -19

Combine like terms: 0 + -19 = -19
-10z + z2 = -19

The z term is -10z.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10z + 25 + z2 = -19 + 25

Reorder the terms:
25 + -10z + z2 = -19 + 25

Combine like terms: -19 + 25 = 6
25 + -10z + z2 = 6

Factor a perfect square on the left side:
(z + -5)(z + -5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(z + -5) equal to 2.449489743 and -2.449489743.


Subproblem 1
z + -5 = 2.449489743

Simplifying
z + -5 = 2.449489743

Reorder the terms:
-5 + z = 2.449489743

Solving
-5 + z = 2.449489743

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '5' to each side of the equation.
-5 + 5 + z = 2.449489743 + 5

Combine like terms: -5 + 5 = 0
0 + z = 2.449489743 + 5
z = 2.449489743 + 5

Combine like terms: 2.449489743 + 5 = 7.449489743
z = 7.449489743

Simplifying
z = 7.449489743


Subproblem 2
z + -5 = -2.449489743

Simplifying
z + -5 = -2.449489743

Reorder the terms:
-5 + z = -2.449489743

Solving
-5 + z = -2.449489743

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '5' to each side of the equation.
-5 + 5 + z = -2.449489743 + 5

Combine like terms: -5 + 5 = 0
0 + z = -2.449489743 + 5
z = -2.449489743 + 5

Combine like terms: -2.449489743 + 5 = 2.550510257
z = 2.550510257

Simplifying
z = 2.550510257


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {7.449489743, 2.550510257}
Solution
z = {7.449489743, 2.550510257}

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