12x^2-33x+16=0

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Solution for 12x^2-33x+16=0 equation:


Simplifying
12x2 + -33x + 16 = 0

Reorder the terms:
16 + -33x + 12x2 = 0

Solving
16 + -33x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
1.333333333 + -2.75x + x2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -2.75x + -1.333333333 + x2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -2.75x + x2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -2.75x + x2 = 0 + -1.333333333
-2.75x + x2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-2.75x + x2 = -1.333333333

The x term is -2.75x.  Take half its coefficient (-1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
-2.75x + 1.890625 + x2 = -1.333333333 + 1.890625

Reorder the terms:
1.890625 + -2.75x + x2 = -1.333333333 + 1.890625

Combine like terms: -1.333333333 + 1.890625 = 0.557291667
1.890625 + -2.75x + x2 = 0.557291667

Factor a perfect square on the left side:
(x + -1.375)(x + -1.375) = 0.557291667

Calculate the square root of the right side: 0.746519703

Break this problem into two subproblems by setting 
(x + -1.375) equal to 0.746519703 and -0.746519703.

Subproblem 1

x + -1.375 = 0.746519703 Simplifying x + -1.375 = 0.746519703 Reorder the terms: -1.375 + x = 0.746519703 Solving -1.375 + x = 0.746519703 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = 0.746519703 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = 0.746519703 + 1.375 x = 0.746519703 + 1.375 Combine like terms: 0.746519703 + 1.375 = 2.121519703 x = 2.121519703 Simplifying x = 2.121519703

Subproblem 2

x + -1.375 = -0.746519703 Simplifying x + -1.375 = -0.746519703 Reorder the terms: -1.375 + x = -0.746519703 Solving -1.375 + x = -0.746519703 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = -0.746519703 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = -0.746519703 + 1.375 x = -0.746519703 + 1.375 Combine like terms: -0.746519703 + 1.375 = 0.628480297 x = 0.628480297 Simplifying x = 0.628480297

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.121519703, 0.628480297}

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