4z^2+6z-3=0

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Solution for 4z^2+6z-3=0 equation:


Simplifying
4z2 + 6z + -3 = 0

Reorder the terms:
-3 + 6z + 4z2 = 0

Solving
-3 + 6z + 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'.
-0.75 + 1.5z + z2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 1.5z + 0.75 + z2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 1.5z + z2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 1.5z + z2 = 0 + 0.75
1.5z + z2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
1.5z + z2 = 0.75

The z term is 1.5z.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5z + 0.5625 + z2 = 0.75 + 0.5625

Reorder the terms:
0.5625 + 1.5z + z2 = 0.75 + 0.5625

Combine like terms: 0.75 + 0.5625 = 1.3125
0.5625 + 1.5z + z2 = 1.3125

Factor a perfect square on the left side:
(z + 0.75)(z + 0.75) = 1.3125

Calculate the square root of the right side: 1.145643924

Break this problem into two subproblems by setting 
(z + 0.75) equal to 1.145643924 and -1.145643924.

Subproblem 1

z + 0.75 = 1.145643924 Simplifying z + 0.75 = 1.145643924 Reorder the terms: 0.75 + z = 1.145643924 Solving 0.75 + z = 1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.145643924 + -0.75 z = 1.145643924 + -0.75 Combine like terms: 1.145643924 + -0.75 = 0.395643924 z = 0.395643924 Simplifying z = 0.395643924

Subproblem 2

z + 0.75 = -1.145643924 Simplifying z + 0.75 = -1.145643924 Reorder the terms: 0.75 + z = -1.145643924 Solving 0.75 + z = -1.145643924 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.145643924 + -0.75 z = -1.145643924 + -0.75 Combine like terms: -1.145643924 + -0.75 = -1.895643924 z = -1.895643924 Simplifying z = -1.895643924

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.395643924, -1.895643924}

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