5x(2x-9)=x^2-5

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Solution for 5x(2x-9)=x^2-5 equation:


Simplifying
5x(2x + -9) = x2 + -5

Reorder the terms:
5x(-9 + 2x) = x2 + -5
(-9 * 5x + 2x * 5x) = x2 + -5
(-45x + 10x2) = x2 + -5

Reorder the terms:
-45x + 10x2 = -5 + x2

Solving
-45x + 10x2 = -5 + x2

Solving for variable 'x'.

Reorder the terms:
5 + -45x + 10x2 + -1x2 = -5 + x2 + 5 + -1x2

Combine like terms: 10x2 + -1x2 = 9x2
5 + -45x + 9x2 = -5 + x2 + 5 + -1x2

Reorder the terms:
5 + -45x + 9x2 = -5 + 5 + x2 + -1x2

Combine like terms: -5 + 5 = 0
5 + -45x + 9x2 = 0 + x2 + -1x2
5 + -45x + 9x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
5 + -45x + 9x2 = 0

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

Divide each side by '9'.
0.5555555556 + -5x + x2 = 0

Move the constant term to the right:

Add '-0.5555555556' to each side of the equation.
0.5555555556 + -5x + -0.5555555556 + x2 = 0 + -0.5555555556

Reorder the terms:
0.5555555556 + -0.5555555556 + -5x + x2 = 0 + -0.5555555556

Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000
0.0000000000 + -5x + x2 = 0 + -0.5555555556
-5x + x2 = 0 + -0.5555555556

Combine like terms: 0 + -0.5555555556 = -0.5555555556
-5x + x2 = -0.5555555556

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = -0.5555555556 + 6.25

Reorder the terms:
6.25 + -5x + x2 = -0.5555555556 + 6.25

Combine like terms: -0.5555555556 + 6.25 = 5.6944444444
6.25 + -5x + x2 = 5.6944444444

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 5.6944444444

Calculate the square root of the right side: 2.386303511

Break this problem into two subproblems by setting 
(x + -2.5) equal to 2.386303511 and -2.386303511.

Subproblem 1

x + -2.5 = 2.386303511 Simplifying x + -2.5 = 2.386303511 Reorder the terms: -2.5 + x = 2.386303511 Solving -2.5 + x = 2.386303511 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = 2.386303511 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = 2.386303511 + 2.5 x = 2.386303511 + 2.5 Combine like terms: 2.386303511 + 2.5 = 4.886303511 x = 4.886303511 Simplifying x = 4.886303511

Subproblem 2

x + -2.5 = -2.386303511 Simplifying x + -2.5 = -2.386303511 Reorder the terms: -2.5 + x = -2.386303511 Solving -2.5 + x = -2.386303511 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + x = -2.386303511 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + x = -2.386303511 + 2.5 x = -2.386303511 + 2.5 Combine like terms: -2.386303511 + 2.5 = 0.113696489 x = 0.113696489 Simplifying x = 0.113696489

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.886303511, 0.113696489}

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