18x+9=2x^2+6

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Solution for 18x+9=2x^2+6 equation:


Simplifying
18x + 9 = 2x2 + 6

Reorder the terms:
9 + 18x = 2x2 + 6

Reorder the terms:
9 + 18x = 6 + 2x2

Solving
9 + 18x = 6 + 2x2

Solving for variable 'x'.

Reorder the terms:
9 + -6 + 18x + -2x2 = 6 + 2x2 + -6 + -2x2

Combine like terms: 9 + -6 = 3
3 + 18x + -2x2 = 6 + 2x2 + -6 + -2x2

Reorder the terms:
3 + 18x + -2x2 = 6 + -6 + 2x2 + -2x2

Combine like terms: 6 + -6 = 0
3 + 18x + -2x2 = 0 + 2x2 + -2x2
3 + 18x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
3 + 18x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-1.5 + -9x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -9x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -9x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -9x + x2 = 0 + 1.5
-9x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-9x + x2 = 1.5

The x term is -9x.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9x + 20.25 + x2 = 1.5 + 20.25

Reorder the terms:
20.25 + -9x + x2 = 1.5 + 20.25

Combine like terms: 1.5 + 20.25 = 21.75
20.25 + -9x + x2 = 21.75

Factor a perfect square on the left side:
(x + -4.5)(x + -4.5) = 21.75

Calculate the square root of the right side: 4.663689527

Break this problem into two subproblems by setting 
(x + -4.5) equal to 4.663689527 and -4.663689527.

Subproblem 1

x + -4.5 = 4.663689527 Simplifying x + -4.5 = 4.663689527 Reorder the terms: -4.5 + x = 4.663689527 Solving -4.5 + x = 4.663689527 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = 4.663689527 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = 4.663689527 + 4.5 x = 4.663689527 + 4.5 Combine like terms: 4.663689527 + 4.5 = 9.163689527 x = 9.163689527 Simplifying x = 9.163689527

Subproblem 2

x + -4.5 = -4.663689527 Simplifying x + -4.5 = -4.663689527 Reorder the terms: -4.5 + x = -4.663689527 Solving -4.5 + x = -4.663689527 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + x = -4.663689527 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + x = -4.663689527 + 4.5 x = -4.663689527 + 4.5 Combine like terms: -4.663689527 + 4.5 = -0.163689527 x = -0.163689527 Simplifying x = -0.163689527

Solution

The solution to the problem is based on the solutions from the subproblems. x = {9.163689527, -0.163689527}

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