3x(3x-1)=x^2+20

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Solution for 3x(3x-1)=x^2+20 equation:


Simplifying
3x(3x + -1) = x2 + 20

Reorder the terms:
3x(-1 + 3x) = x2 + 20
(-1 * 3x + 3x * 3x) = x2 + 20
(-3x + 9x2) = x2 + 20

Reorder the terms:
-3x + 9x2 = 20 + x2

Solving
-3x + 9x2 = 20 + x2

Solving for variable 'x'.

Reorder the terms:
-20 + -3x + 9x2 + -1x2 = 20 + x2 + -20 + -1x2

Combine like terms: 9x2 + -1x2 = 8x2
-20 + -3x + 8x2 = 20 + x2 + -20 + -1x2

Reorder the terms:
-20 + -3x + 8x2 = 20 + -20 + x2 + -1x2

Combine like terms: 20 + -20 = 0
-20 + -3x + 8x2 = 0 + x2 + -1x2
-20 + -3x + 8x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-20 + -3x + 8x2 = 0

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

Divide each side by '8'.
-2.5 + -0.375x + x2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.375x + 2.5 + x2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.375x + x2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.375x + x2 = 0 + 2.5
-0.375x + x2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.375x + x2 = 2.5

The x term is -0.375x.  Take half its coefficient (-0.1875).
Square it (0.03515625) and add it to both sides.

Add '0.03515625' to each side of the equation.
-0.375x + 0.03515625 + x2 = 2.5 + 0.03515625

Reorder the terms:
0.03515625 + -0.375x + x2 = 2.5 + 0.03515625

Combine like terms: 2.5 + 0.03515625 = 2.53515625
0.03515625 + -0.375x + x2 = 2.53515625

Factor a perfect square on the left side:
(x + -0.1875)(x + -0.1875) = 2.53515625

Calculate the square root of the right side: 1.5922174

Break this problem into two subproblems by setting 
(x + -0.1875) equal to 1.5922174 and -1.5922174.

Subproblem 1

x + -0.1875 = 1.5922174 Simplifying x + -0.1875 = 1.5922174 Reorder the terms: -0.1875 + x = 1.5922174 Solving -0.1875 + x = 1.5922174 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = 1.5922174 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = 1.5922174 + 0.1875 x = 1.5922174 + 0.1875 Combine like terms: 1.5922174 + 0.1875 = 1.7797174 x = 1.7797174 Simplifying x = 1.7797174

Subproblem 2

x + -0.1875 = -1.5922174 Simplifying x + -0.1875 = -1.5922174 Reorder the terms: -0.1875 + x = -1.5922174 Solving -0.1875 + x = -1.5922174 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = -1.5922174 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = -1.5922174 + 0.1875 x = -1.5922174 + 0.1875 Combine like terms: -1.5922174 + 0.1875 = -1.4047174 x = -1.4047174 Simplifying x = -1.4047174

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.7797174, -1.4047174}

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