40x^2+x-1.25=0

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Solution for 40x^2+x-1.25=0 equation:


Simplifying
40x2 + x + -1.25 = 0

Reorder the terms:
-1.25 + x + 40x2 = 0

Solving
-1.25 + x + 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'.
-0.03125 + 0.025x + x2 = 0

Move the constant term to the right:

Add '0.03125' to each side of the equation.
-0.03125 + 0.025x + 0.03125 + x2 = 0 + 0.03125

Reorder the terms:
-0.03125 + 0.03125 + 0.025x + x2 = 0 + 0.03125

Combine like terms: -0.03125 + 0.03125 = 0.00000
0.00000 + 0.025x + x2 = 0 + 0.03125
0.025x + x2 = 0 + 0.03125

Combine like terms: 0 + 0.03125 = 0.03125
0.025x + x2 = 0.03125

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.025x + 0.25 + x2 = 0.03125 + 0.25

Reorder the terms:
0.25 + 0.025x + x2 = 0.03125 + 0.25

Combine like terms: 0.03125 + 0.25 = 0.28125
0.25 + 0.025x + x2 = 0.28125

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.28125

Calculate the square root of the right side: 0.530330086

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.530330086 and -0.530330086.

Subproblem 1

x + 0.5 = 0.530330086 Simplifying x + 0.5 = 0.530330086 Reorder the terms: 0.5 + x = 0.530330086 Solving 0.5 + x = 0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.530330086 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.530330086 + -0.5 x = 0.530330086 + -0.5 Combine like terms: 0.530330086 + -0.5 = 0.030330086 x = 0.030330086 Simplifying x = 0.030330086

Subproblem 2

x + 0.5 = -0.530330086 Simplifying x + 0.5 = -0.530330086 Reorder the terms: 0.5 + x = -0.530330086 Solving 0.5 + x = -0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.530330086 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.530330086 + -0.5 x = -0.530330086 + -0.5 Combine like terms: -0.530330086 + -0.5 = -1.030330086 x = -1.030330086 Simplifying x = -1.030330086

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.030330086, -1.030330086}

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