9x+2x^2-24=0

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Solution for 9x+2x^2-24=0 equation:


Simplifying
9x + 2x2 + -24 = 0

Reorder the terms:
-24 + 9x + 2x2 = 0

Solving
-24 + 9x + 2x2 = 0

Solving for variable 'x'.

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

Divide each side by '2'.
-12 + 4.5x + x2 = 0

Move the constant term to the right:

Add '12' to each side of the equation.
-12 + 4.5x + 12 + x2 = 0 + 12

Reorder the terms:
-12 + 12 + 4.5x + x2 = 0 + 12

Combine like terms: -12 + 12 = 0
0 + 4.5x + x2 = 0 + 12
4.5x + x2 = 0 + 12

Combine like terms: 0 + 12 = 12
4.5x + x2 = 12

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 = 12 + 5.0625

Reorder the terms:
5.0625 + 4.5x + x2 = 12 + 5.0625

Combine like terms: 12 + 5.0625 = 17.0625
5.0625 + 4.5x + x2 = 17.0625

Factor a perfect square on the left side:
(x + 2.25)(x + 2.25) = 17.0625

Calculate the square root of the right side: 4.13067791

Break this problem into two subproblems by setting 
(x + 2.25) equal to 4.13067791 and -4.13067791.

Subproblem 1

x + 2.25 = 4.13067791 Simplifying x + 2.25 = 4.13067791 Reorder the terms: 2.25 + x = 4.13067791 Solving 2.25 + x = 4.13067791 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 = 4.13067791 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 4.13067791 + -2.25 x = 4.13067791 + -2.25 Combine like terms: 4.13067791 + -2.25 = 1.88067791 x = 1.88067791 Simplifying x = 1.88067791

Subproblem 2

x + 2.25 = -4.13067791 Simplifying x + 2.25 = -4.13067791 Reorder the terms: 2.25 + x = -4.13067791 Solving 2.25 + x = -4.13067791 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 = -4.13067791 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -4.13067791 + -2.25 x = -4.13067791 + -2.25 Combine like terms: -4.13067791 + -2.25 = -6.38067791 x = -6.38067791 Simplifying x = -6.38067791

Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.88067791, -6.38067791}

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