3x^2-21x+32=0

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Solution for 3x^2-21x+32=0 equation:


Simplifying
3x2 + -21x + 32 = 0

Reorder the terms:
32 + -21x + 3x2 = 0

Solving
32 + -21x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
10.66666667 + -7x + x2 = 0

Move the constant term to the right:

Add '-10.66666667' to each side of the equation.
10.66666667 + -7x + -10.66666667 + x2 = 0 + -10.66666667

Reorder the terms:
10.66666667 + -10.66666667 + -7x + x2 = 0 + -10.66666667

Combine like terms: 10.66666667 + -10.66666667 = 0.00000000
0.00000000 + -7x + x2 = 0 + -10.66666667
-7x + x2 = 0 + -10.66666667

Combine like terms: 0 + -10.66666667 = -10.66666667
-7x + x2 = -10.66666667

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -10.66666667 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -10.66666667 + 12.25

Combine like terms: -10.66666667 + 12.25 = 1.58333333
12.25 + -7x + x2 = 1.58333333

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 1.58333333

Calculate the square root of the right side: 1.258305738

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1.258305738 and -1.258305738.

Subproblem 1

x + -3.5 = 1.258305738 Simplifying x + -3.5 = 1.258305738 Reorder the terms: -3.5 + x = 1.258305738 Solving -3.5 + x = 1.258305738 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 1.258305738 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 1.258305738 + 3.5 x = 1.258305738 + 3.5 Combine like terms: 1.258305738 + 3.5 = 4.758305738 x = 4.758305738 Simplifying x = 4.758305738

Subproblem 2

x + -3.5 = -1.258305738 Simplifying x + -3.5 = -1.258305738 Reorder the terms: -3.5 + x = -1.258305738 Solving -3.5 + x = -1.258305738 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -1.258305738 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -1.258305738 + 3.5 x = -1.258305738 + 3.5 Combine like terms: -1.258305738 + 3.5 = 2.241694262 x = 2.241694262 Simplifying x = 2.241694262

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.758305738, 2.241694262}

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