40x^2-180x-902.5=0

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Solution for 40x^2-180x-902.5=0 equation:


Simplifying
40x2 + -180x + -902.5 = 0

Reorder the terms:
-902.5 + -180x + 40x2 = 0

Solving
-902.5 + -180x + 40x2 = 0

Solving for variable 'x'.

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

Divide each side by '40'.
-22.5625 + -4.5x + x2 = 0

Move the constant term to the right:

Add '22.5625' to each side of the equation.
-22.5625 + -4.5x + 22.5625 + x2 = 0 + 22.5625

Reorder the terms:
-22.5625 + 22.5625 + -4.5x + x2 = 0 + 22.5625

Combine like terms: -22.5625 + 22.5625 = 0.0000
0.0000 + -4.5x + x2 = 0 + 22.5625
-4.5x + x2 = 0 + 22.5625

Combine like terms: 0 + 22.5625 = 22.5625
-4.5x + x2 = 22.5625

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 = 22.5625 + 5.0625

Reorder the terms:
5.0625 + -4.5x + x2 = 22.5625 + 5.0625

Combine like terms: 22.5625 + 5.0625 = 27.625
5.0625 + -4.5x + x2 = 27.625

Factor a perfect square on the left side:
(x + -2.25)(x + -2.25) = 27.625

Calculate the square root of the right side: 5.25594901

Break this problem into two subproblems by setting 
(x + -2.25) equal to 5.25594901 and -5.25594901.

Subproblem 1

x + -2.25 = 5.25594901 Simplifying x + -2.25 = 5.25594901 Reorder the terms: -2.25 + x = 5.25594901 Solving -2.25 + x = 5.25594901 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 = 5.25594901 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 5.25594901 + 2.25 x = 5.25594901 + 2.25 Combine like terms: 5.25594901 + 2.25 = 7.50594901 x = 7.50594901 Simplifying x = 7.50594901

Subproblem 2

x + -2.25 = -5.25594901 Simplifying x + -2.25 = -5.25594901 Reorder the terms: -2.25 + x = -5.25594901 Solving -2.25 + x = -5.25594901 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 = -5.25594901 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -5.25594901 + 2.25 x = -5.25594901 + 2.25 Combine like terms: -5.25594901 + 2.25 = -3.00594901 x = -3.00594901 Simplifying x = -3.00594901

Solution

The solution to the problem is based on the solutions from the subproblems. x = {7.50594901, -3.00594901}

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