2*x^2=9*x+4

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Solution for 2*x^2=9*x+4 equation:


Simplifying
2x2 = 9x + 4

Reorder the terms:
2x2 = 4 + 9x

Solving
2x2 = 4 + 9x

Solving for variable 'x'.

Reorder the terms:
-4 + -9x + 2x2 = 4 + 9x + -4 + -9x

Reorder the terms:
-4 + -9x + 2x2 = 4 + -4 + 9x + -9x

Combine like terms: 4 + -4 = 0
-4 + -9x + 2x2 = 0 + 9x + -9x
-4 + -9x + 2x2 = 9x + -9x

Combine like terms: 9x + -9x = 0
-4 + -9x + 2x2 = 0

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

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

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -4.5x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -4.5x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -4.5x + x2 = 0 + 2
-4.5x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-4.5x + x2 = 2

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 = 2 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 2 + 5.0625

Combine like terms: 2 + 5.0625 = 7.0625
5.0625 + -4.5x + x2 = 7.0625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 7.0625

Calculate the square root of the right side: 2.657536453

Break this problem into two subproblems by setting 
(x + -2.25) equal to 2.657536453 and -2.657536453.

Subproblem 1

x + -2.25 = 2.657536453 Simplifying x + -2.25 = 2.657536453 Reorder the terms: -2.25 + x = 2.657536453 Solving -2.25 + x = 2.657536453 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 = 2.657536453 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 2.657536453 + 2.25 x = 2.657536453 + 2.25 Combine like terms: 2.657536453 + 2.25 = 4.907536453 x = 4.907536453 Simplifying x = 4.907536453

Subproblem 2

x + -2.25 = -2.657536453 Simplifying x + -2.25 = -2.657536453 Reorder the terms: -2.25 + x = -2.657536453 Solving -2.25 + x = -2.657536453 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 = -2.657536453 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -2.657536453 + 2.25 x = -2.657536453 + 2.25 Combine like terms: -2.657536453 + 2.25 = -0.407536453 x = -0.407536453 Simplifying x = -0.407536453

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.907536453, -0.407536453}

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