2a^2-2a-3=a^2+6a

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Solution for 2a^2-2a-3=a^2+6a equation:


Simplifying
2a2 + -2a + -3 = a2 + 6a

Reorder the terms:
-3 + -2a + 2a2 = a2 + 6a

Reorder the terms:
-3 + -2a + 2a2 = 6a + a2

Solving
-3 + -2a + 2a2 = 6a + a2

Solving for variable 'a'.

Reorder the terms:
-3 + -2a + -6a + 2a2 + -1a2 = 6a + a2 + -6a + -1a2

Combine like terms: -2a + -6a = -8a
-3 + -8a + 2a2 + -1a2 = 6a + a2 + -6a + -1a2

Combine like terms: 2a2 + -1a2 = 1a2
-3 + -8a + 1a2 = 6a + a2 + -6a + -1a2

Reorder the terms:
-3 + -8a + 1a2 = 6a + -6a + a2 + -1a2

Combine like terms: 6a + -6a = 0
-3 + -8a + 1a2 = 0 + a2 + -1a2
-3 + -8a + 1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
-3 + -8a + 1a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -8a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + -8a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -8a + a2 = 0 + 3
-8a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
-8a + a2 = 3

The a term is -8a.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8a + 16 + a2 = 3 + 16

Reorder the terms:
16 + -8a + a2 = 3 + 16

Combine like terms: 3 + 16 = 19
16 + -8a + a2 = 19

Factor a perfect square on the left side:
(a + -4)(a + -4) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(a + -4) equal to 4.358898944 and -4.358898944.

Subproblem 1

a + -4 = 4.358898944 Simplifying a + -4 = 4.358898944 Reorder the terms: -4 + a = 4.358898944 Solving -4 + a = 4.358898944 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + a = 4.358898944 + 4 Combine like terms: -4 + 4 = 0 0 + a = 4.358898944 + 4 a = 4.358898944 + 4 Combine like terms: 4.358898944 + 4 = 8.358898944 a = 8.358898944 Simplifying a = 8.358898944

Subproblem 2

a + -4 = -4.358898944 Simplifying a + -4 = -4.358898944 Reorder the terms: -4 + a = -4.358898944 Solving -4 + a = -4.358898944 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + a = -4.358898944 + 4 Combine like terms: -4 + 4 = 0 0 + a = -4.358898944 + 4 a = -4.358898944 + 4 Combine like terms: -4.358898944 + 4 = -0.358898944 a = -0.358898944 Simplifying a = -0.358898944

Solution

The solution to the problem is based on the solutions from the subproblems. a = {8.358898944, -0.358898944}

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