1000=2x^2+9x+4

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Solution for 1000=2x^2+9x+4 equation:


Simplifying
1000 = 2x2 + 9x + 4

Reorder the terms:
1000 = 4 + 9x + 2x2

Solving
1000 = 4 + 9x + 2x2

Solving for variable 'x'.

Combine like terms: 1000 + -4 = 996
996 + -9x + -2x2 = 4 + 9x + 2x2 + -4 + -9x + -2x2

Reorder the terms:
996 + -9x + -2x2 = 4 + -4 + 9x + -9x + 2x2 + -2x2

Combine like terms: 4 + -4 = 0
996 + -9x + -2x2 = 0 + 9x + -9x + 2x2 + -2x2
996 + -9x + -2x2 = 9x + -9x + 2x2 + -2x2

Combine like terms: 9x + -9x = 0
996 + -9x + -2x2 = 0 + 2x2 + -2x2
996 + -9x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
996 + -9x + -2x2 = 0

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

Divide each side by '-2'.
-498 + 4.5x + x2 = 0

Move the constant term to the right:

Add '498' to each side of the equation.
-498 + 4.5x + 498 + x2 = 0 + 498

Reorder the terms:
-498 + 498 + 4.5x + x2 = 0 + 498

Combine like terms: -498 + 498 = 0
0 + 4.5x + x2 = 0 + 498
4.5x + x2 = 0 + 498

Combine like terms: 0 + 498 = 498
4.5x + x2 = 498

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 = 498 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 498 + 5.0625

Combine like terms: 498 + 5.0625 = 503.0625
5.0625 + 4.5x + x2 = 503.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 503.0625

Calculate the square root of the right side: 22.429054817

Break this problem into two subproblems by setting 
(x + 2.25) equal to 22.429054817 and -22.429054817.

Subproblem 1

x + 2.25 = 22.429054817 Simplifying x + 2.25 = 22.429054817 Reorder the terms: 2.25 + x = 22.429054817 Solving 2.25 + x = 22.429054817 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 = 22.429054817 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 22.429054817 + -2.25 x = 22.429054817 + -2.25 Combine like terms: 22.429054817 + -2.25 = 20.179054817 x = 20.179054817 Simplifying x = 20.179054817

Subproblem 2

x + 2.25 = -22.429054817 Simplifying x + 2.25 = -22.429054817 Reorder the terms: 2.25 + x = -22.429054817 Solving 2.25 + x = -22.429054817 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 = -22.429054817 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -22.429054817 + -2.25 x = -22.429054817 + -2.25 Combine like terms: -22.429054817 + -2.25 = -24.679054817 x = -24.679054817 Simplifying x = -24.679054817

Solution

The solution to the problem is based on the solutions from the subproblems. x = {20.179054817, -24.679054817}

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