4z^2+z-6=0

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


Simplifying
4z2 + z + -6 = 0

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

Solving
-6 + z + 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 + 0.25z + z2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + 0.25z + 1.5 + z2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + 0.25z + z2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + 0.25z + z2 = 0 + 1.5
0.25z + z2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
0.25z + z2 = 1.5

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25z + 0.25 + z2 = 1.5 + 0.25

Reorder the terms:
0.25 + 0.25z + z2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + 0.25z + z2 = 1.75

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(z + 0.5) equal to 1.322875656 and -1.322875656.

Subproblem 1

z + 0.5 = 1.322875656 Simplifying z + 0.5 = 1.322875656 Reorder the terms: 0.5 + z = 1.322875656 Solving 0.5 + z = 1.322875656 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.322875656 + -0.5 z = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 z = 0.822875656 Simplifying z = 0.822875656

Subproblem 2

z + 0.5 = -1.322875656 Simplifying z + 0.5 = -1.322875656 Reorder the terms: 0.5 + z = -1.322875656 Solving 0.5 + z = -1.322875656 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.322875656 + -0.5 z = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 z = -1.822875656 Simplifying z = -1.822875656

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.822875656, -1.822875656}

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