7a^2-5a-1=0

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Solution for 7a^2-5a-1=0 equation:


Simplifying
7a2 + -5a + -1 = 0

Reorder the terms:
-1 + -5a + 7a2 = 0

Solving
-1 + -5a + 7a2 = 0

Solving for variable 'a'.

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

Divide each side by '7'.
-0.1428571429 + -0.7142857143a + a2 = 0

Move the constant term to the right:

Add '0.1428571429' to each side of the equation.
-0.1428571429 + -0.7142857143a + 0.1428571429 + a2 = 0 + 0.1428571429

Reorder the terms:
-0.1428571429 + 0.1428571429 + -0.7142857143a + a2 = 0 + 0.1428571429

Combine like terms: -0.1428571429 + 0.1428571429 = 0.0000000000
0.0000000000 + -0.7142857143a + a2 = 0 + 0.1428571429
-0.7142857143a + a2 = 0 + 0.1428571429

Combine like terms: 0 + 0.1428571429 = 0.1428571429
-0.7142857143a + a2 = 0.1428571429

The a term is -0.7142857143a.  Take half its coefficient (-0.3571428572).
Square it (0.1275510204) and add it to both sides.

Add '0.1275510204' to each side of the equation.
-0.7142857143a + 0.1275510204 + a2 = 0.1428571429 + 0.1275510204

Reorder the terms:
0.1275510204 + -0.7142857143a + a2 = 0.1428571429 + 0.1275510204

Combine like terms: 0.1428571429 + 0.1275510204 = 0.2704081633
0.1275510204 + -0.7142857143a + a2 = 0.2704081633

Factor a perfect square on the left side:
(a + -0.3571428572)(a + -0.3571428572) = 0.2704081633

Calculate the square root of the right side: 0.520007849

Break this problem into two subproblems by setting 
(a + -0.3571428572) equal to 0.520007849 and -0.520007849.

Subproblem 1

a + -0.3571428572 = 0.520007849 Simplifying a + -0.3571428572 = 0.520007849 Reorder the terms: -0.3571428572 + a = 0.520007849 Solving -0.3571428572 + a = 0.520007849 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + a = 0.520007849 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + a = 0.520007849 + 0.3571428572 a = 0.520007849 + 0.3571428572 Combine like terms: 0.520007849 + 0.3571428572 = 0.8771507062 a = 0.8771507062 Simplifying a = 0.8771507062

Subproblem 2

a + -0.3571428572 = -0.520007849 Simplifying a + -0.3571428572 = -0.520007849 Reorder the terms: -0.3571428572 + a = -0.520007849 Solving -0.3571428572 + a = -0.520007849 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + a = -0.520007849 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + a = -0.520007849 + 0.3571428572 a = -0.520007849 + 0.3571428572 Combine like terms: -0.520007849 + 0.3571428572 = -0.1628649918 a = -0.1628649918 Simplifying a = -0.1628649918

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.8771507062, -0.1628649918}

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