9x^2=-10x+20

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Solution for 9x^2=-10x+20 equation:


Simplifying
9x2 = -10x + 20

Reorder the terms:
9x2 = 20 + -10x

Solving
9x2 = 20 + -10x

Solving for variable 'x'.

Reorder the terms:
-20 + 10x + 9x2 = 20 + -10x + -20 + 10x

Reorder the terms:
-20 + 10x + 9x2 = 20 + -20 + -10x + 10x

Combine like terms: 20 + -20 = 0
-20 + 10x + 9x2 = 0 + -10x + 10x
-20 + 10x + 9x2 = -10x + 10x

Combine like terms: -10x + 10x = 0
-20 + 10x + 9x2 = 0

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

Divide each side by '9'.
-2.222222222 + 1.111111111x + x2 = 0

Move the constant term to the right:

Add '2.222222222' to each side of the equation.
-2.222222222 + 1.111111111x + 2.222222222 + x2 = 0 + 2.222222222

Reorder the terms:
-2.222222222 + 2.222222222 + 1.111111111x + x2 = 0 + 2.222222222

Combine like terms: -2.222222222 + 2.222222222 = 0.000000000
0.000000000 + 1.111111111x + x2 = 0 + 2.222222222
1.111111111x + x2 = 0 + 2.222222222

Combine like terms: 0 + 2.222222222 = 2.222222222
1.111111111x + x2 = 2.222222222

The x term is 1.111111111x.  Take half its coefficient (0.5555555555).
Square it (0.3086419752) and add it to both sides.

Add '0.3086419752' to each side of the equation.
1.111111111x + 0.3086419752 + x2 = 2.222222222 + 0.3086419752

Reorder the terms:
0.3086419752 + 1.111111111x + x2 = 2.222222222 + 0.3086419752

Combine like terms: 2.222222222 + 0.3086419752 = 2.5308641972
0.3086419752 + 1.111111111x + x2 = 2.5308641972

Factor a perfect square on the left side:
(x + 0.5555555555)(x + 0.5555555555) = 2.5308641972

Calculate the square root of the right side: 1.590869007

Break this problem into two subproblems by setting 
(x + 0.5555555555) equal to 1.590869007 and -1.590869007.

Subproblem 1

x + 0.5555555555 = 1.590869007 Simplifying x + 0.5555555555 = 1.590869007 Reorder the terms: 0.5555555555 + x = 1.590869007 Solving 0.5555555555 + x = 1.590869007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + x = 1.590869007 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + x = 1.590869007 + -0.5555555555 x = 1.590869007 + -0.5555555555 Combine like terms: 1.590869007 + -0.5555555555 = 1.0353134515 x = 1.0353134515 Simplifying x = 1.0353134515

Subproblem 2

x + 0.5555555555 = -1.590869007 Simplifying x + 0.5555555555 = -1.590869007 Reorder the terms: 0.5555555555 + x = -1.590869007 Solving 0.5555555555 + x = -1.590869007 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5555555555' to each side of the equation. 0.5555555555 + -0.5555555555 + x = -1.590869007 + -0.5555555555 Combine like terms: 0.5555555555 + -0.5555555555 = 0.0000000000 0.0000000000 + x = -1.590869007 + -0.5555555555 x = -1.590869007 + -0.5555555555 Combine like terms: -1.590869007 + -0.5555555555 = -2.1464245625 x = -2.1464245625 Simplifying x = -2.1464245625

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.0353134515, -2.1464245625}

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