40x^2-125x-150=0

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Solution for 40x^2-125x-150=0 equation: 


Simplifying
40x2 + -125x + -150 = 0

Reorder the terms:
-150 + -125x + 40x2 = 0

Solving
-150 + -125x + 40x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '5'.
5(-30 + -25x + 8x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-30 + -25x + 8x2)' equal to zero and attempt to solve:

Simplifying
-30 + -25x + 8x2 = 0

Solving
-30 + -25x + 8x2 = 0

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

Divide each side by '8'.
-3.75 + -3.125x + x2 = 0

Move the constant term to the right:

Add '3.75' to each side of the equation.
-3.75 + -3.125x + 3.75 + x2 = 0 + 3.75

Reorder the terms:
-3.75 + 3.75 + -3.125x + x2 = 0 + 3.75

Combine like terms: -3.75 + 3.75 = 0.00
0.00 + -3.125x + x2 = 0 + 3.75
-3.125x + x2 = 0 + 3.75

Combine like terms: 0 + 3.75 = 3.75
-3.125x + x2 = 3.75

The x term is -3.125x.  Take half its coefficient (-1.5625).
Square it (2.44140625) and add it to both sides.

Add '2.44140625' to each side of the equation.
-3.125x + 2.44140625 + x2 = 3.75 + 2.44140625

Reorder the terms:
2.44140625 + -3.125x + x2 = 3.75 + 2.44140625

Combine like terms: 3.75 + 2.44140625 = 6.19140625
2.44140625 + -3.125x + x2 = 6.19140625

Factor a perfect square on the left side:
(x + -1.5625)(x + -1.5625) = 6.19140625

Calculate the square root of the right side: 2.488253655

Break this problem into two subproblems by setting 
(x + -1.5625) equal to 2.488253655 and -2.488253655.


Subproblem 1
x + -1.5625 = 2.488253655

Simplifying
x + -1.5625 = 2.488253655

Reorder the terms:
-1.5625 + x = 2.488253655

Solving
-1.5625 + x = 2.488253655

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.5625' to each side of the equation.
-1.5625 + 1.5625 + x = 2.488253655 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + x = 2.488253655 + 1.5625
x = 2.488253655 + 1.5625

Combine like terms: 2.488253655 + 1.5625 = 4.050753655
x = 4.050753655

Simplifying
x = 4.050753655


Subproblem 2
x + -1.5625 = -2.488253655

Simplifying
x + -1.5625 = -2.488253655

Reorder the terms:
-1.5625 + x = -2.488253655

Solving
-1.5625 + x = -2.488253655

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '1.5625' to each side of the equation.
-1.5625 + 1.5625 + x = -2.488253655 + 1.5625

Combine like terms: -1.5625 + 1.5625 = 0.0000
0.0000 + x = -2.488253655 + 1.5625
x = -2.488253655 + 1.5625

Combine like terms: -2.488253655 + 1.5625 = -0.925753655
x = -0.925753655

Simplifying
x = -0.925753655


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.050753655, -0.925753655}
Solution
x = {4.050753655, -0.925753655}

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