2x^2+24x+45=0

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Solution for 2x^2+24x+45=0 equation:


Simplifying
2x2 + 24x + 45 = 0

Reorder the terms:
45 + 24x + 2x2 = 0

Solving
45 + 24x + 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'.
22.5 + 12x + x2 = 0

Move the constant term to the right:

Add '-22.5' to each side of the equation.
22.5 + 12x + -22.5 + x2 = 0 + -22.5

Reorder the terms:
22.5 + -22.5 + 12x + x2 = 0 + -22.5

Combine like terms: 22.5 + -22.5 = 0.0
0.0 + 12x + x2 = 0 + -22.5
12x + x2 = 0 + -22.5

Combine like terms: 0 + -22.5 = -22.5
12x + x2 = -22.5

The x term is 12x.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12x + 36 + x2 = -22.5 + 36

Reorder the terms:
36 + 12x + x2 = -22.5 + 36

Combine like terms: -22.5 + 36 = 13.5
36 + 12x + x2 = 13.5

Factor a perfect square on the left side:
(x + 6)(x + 6) = 13.5

Calculate the square root of the right side: 3.674234614

Break this problem into two subproblems by setting 
(x + 6) equal to 3.674234614 and -3.674234614.

Subproblem 1

x + 6 = 3.674234614 Simplifying x + 6 = 3.674234614 Reorder the terms: 6 + x = 3.674234614 Solving 6 + x = 3.674234614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = 3.674234614 + -6 Combine like terms: 6 + -6 = 0 0 + x = 3.674234614 + -6 x = 3.674234614 + -6 Combine like terms: 3.674234614 + -6 = -2.325765386 x = -2.325765386 Simplifying x = -2.325765386

Subproblem 2

x + 6 = -3.674234614 Simplifying x + 6 = -3.674234614 Reorder the terms: 6 + x = -3.674234614 Solving 6 + x = -3.674234614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + x = -3.674234614 + -6 Combine like terms: 6 + -6 = 0 0 + x = -3.674234614 + -6 x = -3.674234614 + -6 Combine like terms: -3.674234614 + -6 = -9.674234614 x = -9.674234614 Simplifying x = -9.674234614

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-2.325765386, -9.674234614}

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