4a^2+9a-1=0

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Solution for 4a^2+9a-1=0 equation:


Simplifying
4a2 + 9a + -1 = 0

Reorder the terms:
-1 + 9a + 4a2 = 0

Solving
-1 + 9a + 4a2 = 0

Solving for variable 'a'.

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

Divide each side by '4'.
-0.25 + 2.25a + a2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 2.25a + 0.25 + a2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 2.25a + a2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 2.25a + a2 = 0 + 0.25
2.25a + a2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
2.25a + a2 = 0.25

The a term is 2.25a.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25a + 1.265625 + a2 = 0.25 + 1.265625

Reorder the terms:
1.265625 + 2.25a + a2 = 0.25 + 1.265625

Combine like terms: 0.25 + 1.265625 = 1.515625
1.265625 + 2.25a + a2 = 1.515625

Factor a perfect square on the left side:
(a + 1.125)(a + 1.125) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(a + 1.125) equal to 1.231107225 and -1.231107225.

Subproblem 1

a + 1.125 = 1.231107225 Simplifying a + 1.125 = 1.231107225 Reorder the terms: 1.125 + a = 1.231107225 Solving 1.125 + a = 1.231107225 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + a = 1.231107225 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + a = 1.231107225 + -1.125 a = 1.231107225 + -1.125 Combine like terms: 1.231107225 + -1.125 = 0.106107225 a = 0.106107225 Simplifying a = 0.106107225

Subproblem 2

a + 1.125 = -1.231107225 Simplifying a + 1.125 = -1.231107225 Reorder the terms: 1.125 + a = -1.231107225 Solving 1.125 + a = -1.231107225 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.125' to each side of the equation. 1.125 + -1.125 + a = -1.231107225 + -1.125 Combine like terms: 1.125 + -1.125 = 0.000 0.000 + a = -1.231107225 + -1.125 a = -1.231107225 + -1.125 Combine like terms: -1.231107225 + -1.125 = -2.356107225 a = -2.356107225 Simplifying a = -2.356107225

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.106107225, -2.356107225}

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