3x^2-30x+61=0

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Solution for 3x^2-30x+61=0 equation:


Simplifying
3x2 + -30x + 61 = 0

Reorder the terms:
61 + -30x + 3x2 = 0

Solving
61 + -30x + 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'.
20.33333333 + -10x + x2 = 0

Move the constant term to the right:

Add '-20.33333333' to each side of the equation.
20.33333333 + -10x + -20.33333333 + x2 = 0 + -20.33333333

Reorder the terms:
20.33333333 + -20.33333333 + -10x + x2 = 0 + -20.33333333

Combine like terms: 20.33333333 + -20.33333333 = 0.00000000
0.00000000 + -10x + x2 = 0 + -20.33333333
-10x + x2 = 0 + -20.33333333

Combine like terms: 0 + -20.33333333 = -20.33333333
-10x + x2 = -20.33333333

The x term is -10x.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10x + 25 + x2 = -20.33333333 + 25

Reorder the terms:
25 + -10x + x2 = -20.33333333 + 25

Combine like terms: -20.33333333 + 25 = 4.66666667
25 + -10x + x2 = 4.66666667

Factor a perfect square on the left side:
(x + -5)(x + -5) = 4.66666667

Calculate the square root of the right side: 2.1602469

Break this problem into two subproblems by setting 
(x + -5) equal to 2.1602469 and -2.1602469.

Subproblem 1

x + -5 = 2.1602469 Simplifying x + -5 = 2.1602469 Reorder the terms: -5 + x = 2.1602469 Solving -5 + x = 2.1602469 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = 2.1602469 + 5 Combine like terms: -5 + 5 = 0 0 + x = 2.1602469 + 5 x = 2.1602469 + 5 Combine like terms: 2.1602469 + 5 = 7.1602469 x = 7.1602469 Simplifying x = 7.1602469

Subproblem 2

x + -5 = -2.1602469 Simplifying x + -5 = -2.1602469 Reorder the terms: -5 + x = -2.1602469 Solving -5 + x = -2.1602469 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + x = -2.1602469 + 5 Combine like terms: -5 + 5 = 0 0 + x = -2.1602469 + 5 x = -2.1602469 + 5 Combine like terms: -2.1602469 + 5 = 2.8397531 x = 2.8397531 Simplifying x = 2.8397531

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.1602469, 2.8397531}

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