a(3a+1)+(a+1)(a-1)=

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Solution for a(3a+1)+(a+1)(a-1)= equation:


Simplifying
a(3a + 1) + (a + 1)(a + -1) = 0

Reorder the terms:
a(1 + 3a) + (a + 1)(a + -1) = 0
(1 * a + 3a * a) + (a + 1)(a + -1) = 0
(1a + 3a2) + (a + 1)(a + -1) = 0

Reorder the terms:
1a + 3a2 + (1 + a)(a + -1) = 0

Reorder the terms:
1a + 3a2 + (1 + a)(-1 + a) = 0

Multiply (1 + a) * (-1 + a)
1a + 3a2 + (1(-1 + a) + a(-1 + a)) = 0
1a + 3a2 + ((-1 * 1 + a * 1) + a(-1 + a)) = 0
1a + 3a2 + ((-1 + 1a) + a(-1 + a)) = 0
1a + 3a2 + (-1 + 1a + (-1 * a + a * a)) = 0
1a + 3a2 + (-1 + 1a + (-1a + a2)) = 0

Combine like terms: 1a + -1a = 0
1a + 3a2 + (-1 + 0 + a2) = 0
1a + 3a2 + (-1 + a2) = 0

Reorder the terms:
-1 + 1a + 3a2 + a2 = 0

Combine like terms: 3a2 + a2 = 4a2
-1 + 1a + 4a2 = 0

Solving
-1 + 1a + 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 + 0.25a + a2 = 0

Move the constant term to the right:

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

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

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

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

The a term is 0.25a.  Take half its coefficient (0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
0.25a + 0.015625 + a2 = 0.25 + 0.015625

Reorder the terms:
0.015625 + 0.25a + a2 = 0.25 + 0.015625

Combine like terms: 0.25 + 0.015625 = 0.265625
0.015625 + 0.25a + a2 = 0.265625

Factor a perfect square on the left side:
(a + 0.125)(a + 0.125) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(a + 0.125) equal to 0.515388203 and -0.515388203.

Subproblem 1

a + 0.125 = 0.515388203 Simplifying a + 0.125 = 0.515388203 Reorder the terms: 0.125 + a = 0.515388203 Solving 0.125 + a = 0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = 0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = 0.515388203 + -0.125 a = 0.515388203 + -0.125 Combine like terms: 0.515388203 + -0.125 = 0.390388203 a = 0.390388203 Simplifying a = 0.390388203

Subproblem 2

a + 0.125 = -0.515388203 Simplifying a + 0.125 = -0.515388203 Reorder the terms: 0.125 + a = -0.515388203 Solving 0.125 + a = -0.515388203 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = -0.515388203 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = -0.515388203 + -0.125 a = -0.515388203 + -0.125 Combine like terms: -0.515388203 + -0.125 = -0.640388203 a = -0.640388203 Simplifying a = -0.640388203

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.390388203, -0.640388203}

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