9x^2=30x-20

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Solution for 9x^2=30x-20 equation:


Simplifying
9x2 = 30x + -20

Reorder the terms:
9x2 = -20 + 30x

Solving
9x2 = -20 + 30x

Solving for variable 'x'.

Reorder the terms:
20 + -30x + 9x2 = -20 + 30x + 20 + -30x

Reorder the terms:
20 + -30x + 9x2 = -20 + 20 + 30x + -30x

Combine like terms: -20 + 20 = 0
20 + -30x + 9x2 = 0 + 30x + -30x
20 + -30x + 9x2 = 30x + -30x

Combine like terms: 30x + -30x = 0
20 + -30x + 9x2 = 0

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

Divide each side by '9'.
2.222222222 + -3.333333333x + x2 = 0

Move the constant term to the right:

Add '-2.222222222' to each side of the equation.
2.222222222 + -3.333333333x + -2.222222222 + x2 = 0 + -2.222222222

Reorder the terms:
2.222222222 + -2.222222222 + -3.333333333x + x2 = 0 + -2.222222222

Combine like terms: 2.222222222 + -2.222222222 = 0.000000000
0.000000000 + -3.333333333x + x2 = 0 + -2.222222222
-3.333333333x + x2 = 0 + -2.222222222

Combine like terms: 0 + -2.222222222 = -2.222222222
-3.333333333x + x2 = -2.222222222

The x term is -3.333333333x.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333x + 2.777777779 + x2 = -2.222222222 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333x + x2 = -2.222222222 + 2.777777779

Combine like terms: -2.222222222 + 2.777777779 = 0.555555557
2.777777779 + -3.333333333x + x2 = 0.555555557

Factor a perfect square on the left side:
(x + -1.666666667)(x + -1.666666667) = 0.555555557

Calculate the square root of the right side: 0.745355993

Break this problem into two subproblems by setting 
(x + -1.666666667) equal to 0.745355993 and -0.745355993.

Subproblem 1

x + -1.666666667 = 0.745355993 Simplifying x + -1.666666667 = 0.745355993 Reorder the terms: -1.666666667 + x = 0.745355993 Solving -1.666666667 + x = 0.745355993 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = 0.745355993 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = 0.745355993 + 1.666666667 x = 0.745355993 + 1.666666667 Combine like terms: 0.745355993 + 1.666666667 = 2.41202266 x = 2.41202266 Simplifying x = 2.41202266

Subproblem 2

x + -1.666666667 = -0.745355993 Simplifying x + -1.666666667 = -0.745355993 Reorder the terms: -1.666666667 + x = -0.745355993 Solving -1.666666667 + x = -0.745355993 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.666666667' to each side of the equation. -1.666666667 + 1.666666667 + x = -0.745355993 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + x = -0.745355993 + 1.666666667 x = -0.745355993 + 1.666666667 Combine like terms: -0.745355993 + 1.666666667 = 0.921310674 x = 0.921310674 Simplifying x = 0.921310674

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.41202266, 0.921310674}

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