18x^2+31x-40=0

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Solution for 18x^2+31x-40=0 equation:


Simplifying
18x2 + 31x + -40 = 0

Reorder the terms:
-40 + 31x + 18x2 = 0

Solving
-40 + 31x + 18x2 = 0

Solving for variable 'x'.

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

Divide each side by '18'.
-2.222222222 + 1.722222222x + x2 = 0

Move the constant term to the right:

Add '2.222222222' to each side of the equation.
-2.222222222 + 1.722222222x + 2.222222222 + x2 = 0 + 2.222222222

Reorder the terms:
-2.222222222 + 2.222222222 + 1.722222222x + x2 = 0 + 2.222222222

Combine like terms: -2.222222222 + 2.222222222 = 0.000000000
0.000000000 + 1.722222222x + x2 = 0 + 2.222222222
1.722222222x + x2 = 0 + 2.222222222

Combine like terms: 0 + 2.222222222 = 2.222222222
1.722222222x + x2 = 2.222222222

The x term is 1.722222222x.  Take half its coefficient (0.861111111).
Square it (0.7415123455) and add it to both sides.

Add '0.7415123455' to each side of the equation.
1.722222222x + 0.7415123455 + x2 = 2.222222222 + 0.7415123455

Reorder the terms:
0.7415123455 + 1.722222222x + x2 = 2.222222222 + 0.7415123455

Combine like terms: 2.222222222 + 0.7415123455 = 2.9637345675
0.7415123455 + 1.722222222x + x2 = 2.9637345675

Factor a perfect square on the left side:
(x + 0.861111111)(x + 0.861111111) = 2.9637345675

Calculate the square root of the right side: 1.721550048

Break this problem into two subproblems by setting 
(x + 0.861111111) equal to 1.721550048 and -1.721550048.

Subproblem 1

x + 0.861111111 = 1.721550048 Simplifying x + 0.861111111 = 1.721550048 Reorder the terms: 0.861111111 + x = 1.721550048 Solving 0.861111111 + x = 1.721550048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.861111111' to each side of the equation. 0.861111111 + -0.861111111 + x = 1.721550048 + -0.861111111 Combine like terms: 0.861111111 + -0.861111111 = 0.000000000 0.000000000 + x = 1.721550048 + -0.861111111 x = 1.721550048 + -0.861111111 Combine like terms: 1.721550048 + -0.861111111 = 0.860438937 x = 0.860438937 Simplifying x = 0.860438937

Subproblem 2

x + 0.861111111 = -1.721550048 Simplifying x + 0.861111111 = -1.721550048 Reorder the terms: 0.861111111 + x = -1.721550048 Solving 0.861111111 + x = -1.721550048 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.861111111' to each side of the equation. 0.861111111 + -0.861111111 + x = -1.721550048 + -0.861111111 Combine like terms: 0.861111111 + -0.861111111 = 0.000000000 0.000000000 + x = -1.721550048 + -0.861111111 x = -1.721550048 + -0.861111111 Combine like terms: -1.721550048 + -0.861111111 = -2.582661159 x = -2.582661159 Simplifying x = -2.582661159

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.860438937, -2.582661159}

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