2x^2=9x+3

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Solution for 2x^2=9x+3 equation:


Simplifying
2x2 = 9x + 3

Reorder the terms:
2x2 = 3 + 9x

Solving
2x2 = 3 + 9x

Solving for variable 'x'.

Reorder the terms:
-3 + -9x + 2x2 = 3 + 9x + -3 + -9x

Reorder the terms:
-3 + -9x + 2x2 = 3 + -3 + 9x + -9x

Combine like terms: 3 + -3 = 0
-3 + -9x + 2x2 = 0 + 9x + -9x
-3 + -9x + 2x2 = 9x + -9x

Combine like terms: 9x + -9x = 0
-3 + -9x + 2x2 = 0

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

Divide each side by '2'.
-1.5 + -4.5x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -4.5x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -4.5x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -4.5x + x2 = 0 + 1.5
-4.5x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-4.5x + x2 = 1.5

The x term is -4.5x.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5x + 5.0625 + x2 = 1.5 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 1.5 + 5.0625

Combine like terms: 1.5 + 5.0625 = 6.5625
5.0625 + -4.5x + x2 = 6.5625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.561737691 and -2.561737691.

Subproblem 1

x + -2.25 = 2.561737691 Simplifying x + -2.25 = 2.561737691 Reorder the terms: -2.25 + x = 2.561737691 Solving -2.25 + x = 2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 2.561737691 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.561737691 + 2.25 x = 2.561737691 + 2.25 Combine like terms: 2.561737691 + 2.25 = 4.811737691 x = 4.811737691 Simplifying x = 4.811737691

Subproblem 2

x + -2.25 = -2.561737691 Simplifying x + -2.25 = -2.561737691 Reorder the terms: -2.25 + x = -2.561737691 Solving -2.25 + x = -2.561737691 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -2.561737691 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.561737691 + 2.25 x = -2.561737691 + 2.25 Combine like terms: -2.561737691 + 2.25 = -0.311737691 x = -0.311737691 Simplifying x = -0.311737691

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.811737691, -0.311737691}

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