2a(a-2)=3a+13

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Solution for 2a(a-2)=3a+13 equation:


Simplifying
2a(a + -2) = 3a + 13

Reorder the terms:
2a(-2 + a) = 3a + 13
(-2 * 2a + a * 2a) = 3a + 13
(-4a + 2a2) = 3a + 13

Reorder the terms:
-4a + 2a2 = 13 + 3a

Solving
-4a + 2a2 = 13 + 3a

Solving for variable 'a'.

Reorder the terms:
-13 + -4a + -3a + 2a2 = 13 + 3a + -13 + -3a

Combine like terms: -4a + -3a = -7a
-13 + -7a + 2a2 = 13 + 3a + -13 + -3a

Reorder the terms:
-13 + -7a + 2a2 = 13 + -13 + 3a + -3a

Combine like terms: 13 + -13 = 0
-13 + -7a + 2a2 = 0 + 3a + -3a
-13 + -7a + 2a2 = 3a + -3a

Combine like terms: 3a + -3a = 0
-13 + -7a + 2a2 = 0

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

Divide each side by '2'.
-6.5 + -3.5a + a2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + -3.5a + 6.5 + a2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + -3.5a + a2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + -3.5a + a2 = 0 + 6.5
-3.5a + a2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
-3.5a + a2 = 6.5

The a term is -3.5a.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5a + 3.0625 + a2 = 6.5 + 3.0625

Reorder the terms:
3.0625 + -3.5a + a2 = 6.5 + 3.0625

Combine like terms: 6.5 + 3.0625 = 9.5625
3.0625 + -3.5a + a2 = 9.5625

Factor a perfect square on the left side:
(a + -1.75)(a + -1.75) = 9.5625

Calculate the square root of the right side: 3.092329219

Break this problem into two subproblems by setting 
(a + -1.75) equal to 3.092329219 and -3.092329219.

Subproblem 1

a + -1.75 = 3.092329219 Simplifying a + -1.75 = 3.092329219 Reorder the terms: -1.75 + a = 3.092329219 Solving -1.75 + a = 3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = 3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = 3.092329219 + 1.75 a = 3.092329219 + 1.75 Combine like terms: 3.092329219 + 1.75 = 4.842329219 a = 4.842329219 Simplifying a = 4.842329219

Subproblem 2

a + -1.75 = -3.092329219 Simplifying a + -1.75 = -3.092329219 Reorder the terms: -1.75 + a = -3.092329219 Solving -1.75 + a = -3.092329219 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + a = -3.092329219 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + a = -3.092329219 + 1.75 a = -3.092329219 + 1.75 Combine like terms: -3.092329219 + 1.75 = -1.342329219 a = -1.342329219 Simplifying a = -1.342329219

Solution

The solution to the problem is based on the solutions from the subproblems. a = {4.842329219, -1.342329219}

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