40a^2-42a-9=0

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Solution for 40a^2-42a-9=0 equation:


Simplifying
40a2 + -42a + -9 = 0

Reorder the terms:
-9 + -42a + 40a2 = 0

Solving
-9 + -42a + 40a2 = 0

Solving for variable 'a'.

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

Divide each side by '40'.
-0.225 + -1.05a + a2 = 0

Move the constant term to the right:

Add '0.225' to each side of the equation.
-0.225 + -1.05a + 0.225 + a2 = 0 + 0.225

Reorder the terms:
-0.225 + 0.225 + -1.05a + a2 = 0 + 0.225

Combine like terms: -0.225 + 0.225 = 0.000
0.000 + -1.05a + a2 = 0 + 0.225
-1.05a + a2 = 0 + 0.225

Combine like terms: 0 + 0.225 = 0.225
-1.05a + a2 = 0.225

The a term is -1.05a.  Take half its coefficient (-0.525).
Square it (0.275625) and add it to both sides.

Add '0.275625' to each side of the equation.
-1.05a + 0.275625 + a2 = 0.225 + 0.275625

Reorder the terms:
0.275625 + -1.05a + a2 = 0.225 + 0.275625

Combine like terms: 0.225 + 0.275625 = 0.500625
0.275625 + -1.05a + a2 = 0.500625

Factor a perfect square on the left side:
(a + -0.525)(a + -0.525) = 0.500625

Calculate the square root of the right side: 0.707548585

Break this problem into two subproblems by setting 
(a + -0.525) equal to 0.707548585 and -0.707548585.

Subproblem 1

a + -0.525 = 0.707548585 Simplifying a + -0.525 = 0.707548585 Reorder the terms: -0.525 + a = 0.707548585 Solving -0.525 + a = 0.707548585 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + a = 0.707548585 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + a = 0.707548585 + 0.525 a = 0.707548585 + 0.525 Combine like terms: 0.707548585 + 0.525 = 1.232548585 a = 1.232548585 Simplifying a = 1.232548585

Subproblem 2

a + -0.525 = -0.707548585 Simplifying a + -0.525 = -0.707548585 Reorder the terms: -0.525 + a = -0.707548585 Solving -0.525 + a = -0.707548585 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.525' to each side of the equation. -0.525 + 0.525 + a = -0.707548585 + 0.525 Combine like terms: -0.525 + 0.525 = 0.000 0.000 + a = -0.707548585 + 0.525 a = -0.707548585 + 0.525 Combine like terms: -0.707548585 + 0.525 = -0.182548585 a = -0.182548585 Simplifying a = -0.182548585

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.232548585, -0.182548585}

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