2x^2-23x=5x^2-7x-9

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Solution for 2x^2-23x=5x^2-7x-9 equation:


Simplifying
2x2 + -23x = 5x2 + -7x + -9

Reorder the terms:
-23x + 2x2 = 5x2 + -7x + -9

Reorder the terms:
-23x + 2x2 = -9 + -7x + 5x2

Solving
-23x + 2x2 = -9 + -7x + 5x2

Solving for variable 'x'.

Reorder the terms:
9 + -23x + 7x + 2x2 + -5x2 = -9 + -7x + 5x2 + 9 + 7x + -5x2

Combine like terms: -23x + 7x = -16x
9 + -16x + 2x2 + -5x2 = -9 + -7x + 5x2 + 9 + 7x + -5x2

Combine like terms: 2x2 + -5x2 = -3x2
9 + -16x + -3x2 = -9 + -7x + 5x2 + 9 + 7x + -5x2

Reorder the terms:
9 + -16x + -3x2 = -9 + 9 + -7x + 7x + 5x2 + -5x2

Combine like terms: -9 + 9 = 0
9 + -16x + -3x2 = 0 + -7x + 7x + 5x2 + -5x2
9 + -16x + -3x2 = -7x + 7x + 5x2 + -5x2

Combine like terms: -7x + 7x = 0
9 + -16x + -3x2 = 0 + 5x2 + -5x2
9 + -16x + -3x2 = 5x2 + -5x2

Combine like terms: 5x2 + -5x2 = 0
9 + -16x + -3x2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
-3 + 5.333333333x + x2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 5.333333333x + 3 + x2 = 0 + 3

Reorder the terms:
-3 + 3 + 5.333333333x + x2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 5.333333333x + x2 = 0 + 3
5.333333333x + x2 = 0 + 3

Combine like terms: 0 + 3 = 3
5.333333333x + x2 = 3

The x term is 5.333333333x.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333x + 7.111111113 + x2 = 3 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333x + x2 = 3 + 7.111111113

Combine like terms: 3 + 7.111111113 = 10.111111113
7.111111113 + 5.333333333x + x2 = 10.111111113

Factor a perfect square on the left side:
(x + 2.666666667)(x + 2.666666667) = 10.111111113

Calculate the square root of the right side: 3.179797338

Break this problem into two subproblems by setting 
(x + 2.666666667) equal to 3.179797338 and -3.179797338.

Subproblem 1

x + 2.666666667 = 3.179797338 Simplifying x + 2.666666667 = 3.179797338 Reorder the terms: 2.666666667 + x = 3.179797338 Solving 2.666666667 + x = 3.179797338 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = 3.179797338 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = 3.179797338 + -2.666666667 x = 3.179797338 + -2.666666667 Combine like terms: 3.179797338 + -2.666666667 = 0.513130671 x = 0.513130671 Simplifying x = 0.513130671

Subproblem 2

x + 2.666666667 = -3.179797338 Simplifying x + 2.666666667 = -3.179797338 Reorder the terms: 2.666666667 + x = -3.179797338 Solving 2.666666667 + x = -3.179797338 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + x = -3.179797338 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + x = -3.179797338 + -2.666666667 x = -3.179797338 + -2.666666667 Combine like terms: -3.179797338 + -2.666666667 = -5.846464005 x = -5.846464005 Simplifying x = -5.846464005

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.513130671, -5.846464005}

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