3x^2-32x+75=0

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Solution for 3x^2-32x+75=0 equation:


Simplifying
3x2 + -32x + 75 = 0

Reorder the terms:
75 + -32x + 3x2 = 0

Solving
75 + -32x + 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'.
25 + -10.66666667x + x2 = 0

Move the constant term to the right:

Add '-25' to each side of the equation.
25 + -10.66666667x + -25 + x2 = 0 + -25

Reorder the terms:
25 + -25 + -10.66666667x + x2 = 0 + -25

Combine like terms: 25 + -25 = 0
0 + -10.66666667x + x2 = 0 + -25
-10.66666667x + x2 = 0 + -25

Combine like terms: 0 + -25 = -25
-10.66666667x + x2 = -25

The x term is -10.66666667x.  Take half its coefficient (-5.333333335).
Square it (28.44444446) and add it to both sides.

Add '28.44444446' to each side of the equation.
-10.66666667x + 28.44444446 + x2 = -25 + 28.44444446

Reorder the terms:
28.44444446 + -10.66666667x + x2 = -25 + 28.44444446

Combine like terms: -25 + 28.44444446 = 3.44444446
28.44444446 + -10.66666667x + x2 = 3.44444446

Factor a perfect square on the left side:
(x + -5.333333335)(x + -5.333333335) = 3.44444446

Calculate the square root of the right side: 1.855921458

Break this problem into two subproblems by setting 
(x + -5.333333335) equal to 1.855921458 and -1.855921458.

Subproblem 1

x + -5.333333335 = 1.855921458 Simplifying x + -5.333333335 = 1.855921458 Reorder the terms: -5.333333335 + x = 1.855921458 Solving -5.333333335 + x = 1.855921458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + x = 1.855921458 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + x = 1.855921458 + 5.333333335 x = 1.855921458 + 5.333333335 Combine like terms: 1.855921458 + 5.333333335 = 7.189254793 x = 7.189254793 Simplifying x = 7.189254793

Subproblem 2

x + -5.333333335 = -1.855921458 Simplifying x + -5.333333335 = -1.855921458 Reorder the terms: -5.333333335 + x = -1.855921458 Solving -5.333333335 + x = -1.855921458 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5.333333335' to each side of the equation. -5.333333335 + 5.333333335 + x = -1.855921458 + 5.333333335 Combine like terms: -5.333333335 + 5.333333335 = 0.000000000 0.000000000 + x = -1.855921458 + 5.333333335 x = -1.855921458 + 5.333333335 Combine like terms: -1.855921458 + 5.333333335 = 3.477411877 x = 3.477411877 Simplifying x = 3.477411877

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.189254793, 3.477411877}

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