a(5a-3)=2(a^2+5)

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Solution for a(5a-3)=2(a^2+5) equation:


Simplifying
a(5a + -3) = 2(a2 + 5)

Reorder the terms:
a(-3 + 5a) = 2(a2 + 5)
(-3 * a + 5a * a) = 2(a2 + 5)
(-3a + 5a2) = 2(a2 + 5)

Reorder the terms:
-3a + 5a2 = 2(5 + a2)
-3a + 5a2 = (5 * 2 + a2 * 2)
-3a + 5a2 = (10 + 2a2)

Solving
-3a + 5a2 = 10 + 2a2

Solving for variable 'a'.

Reorder the terms:
-10 + -3a + 5a2 + -2a2 = 10 + 2a2 + -10 + -2a2

Combine like terms: 5a2 + -2a2 = 3a2
-10 + -3a + 3a2 = 10 + 2a2 + -10 + -2a2

Reorder the terms:
-10 + -3a + 3a2 = 10 + -10 + 2a2 + -2a2

Combine like terms: 10 + -10 = 0
-10 + -3a + 3a2 = 0 + 2a2 + -2a2
-10 + -3a + 3a2 = 2a2 + -2a2

Combine like terms: 2a2 + -2a2 = 0
-10 + -3a + 3a2 = 0

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

Divide each side by '3'.
-3.333333333 + -1a + a2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -1a + 3.333333333 + a2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -1a + a2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -1a + a2 = 0 + 3.333333333
-1a + a2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-1a + a2 = 3.333333333

The a term is -1a.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1a + 0.25 + a2 = 3.333333333 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 3.333333333 + 0.25

Combine like terms: 3.333333333 + 0.25 = 3.583333333
0.25 + -1a + a2 = 3.583333333

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 3.583333333

Calculate the square root of the right side: 1.892969449

Break this problem into two subproblems by setting 
(a + -0.5) equal to 1.892969449 and -1.892969449.

Subproblem 1

a + -0.5 = 1.892969449 Simplifying a + -0.5 = 1.892969449 Reorder the terms: -0.5 + a = 1.892969449 Solving -0.5 + a = 1.892969449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 1.892969449 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 1.892969449 + 0.5 a = 1.892969449 + 0.5 Combine like terms: 1.892969449 + 0.5 = 2.392969449 a = 2.392969449 Simplifying a = 2.392969449

Subproblem 2

a + -0.5 = -1.892969449 Simplifying a + -0.5 = -1.892969449 Reorder the terms: -0.5 + a = -1.892969449 Solving -0.5 + a = -1.892969449 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -1.892969449 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -1.892969449 + 0.5 a = -1.892969449 + 0.5 Combine like terms: -1.892969449 + 0.5 = -1.392969449 a = -1.392969449 Simplifying a = -1.392969449

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.392969449, -1.392969449}

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