(5a+2)-(3a^2+1)=

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Solution for (5a+2)-(3a^2+1)= equation:


Simplifying
(5a + 2) + -1(3a2 + 1) = 0

Reorder the terms:
(2 + 5a) + -1(3a2 + 1) = 0

Remove parenthesis around (2 + 5a)
2 + 5a + -1(3a2 + 1) = 0

Reorder the terms:
2 + 5a + -1(1 + 3a2) = 0
2 + 5a + (1 * -1 + 3a2 * -1) = 0
2 + 5a + (-1 + -3a2) = 0

Reorder the terms:
2 + -1 + 5a + -3a2 = 0

Combine like terms: 2 + -1 = 1
1 + 5a + -3a2 = 0

Solving
1 + 5a + -3a2 = 0

Solving for variable 'a'.

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

Divide each side by '-3'.
-0.3333333333 + -1.666666667a + a2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + -1.666666667a + 0.3333333333 + a2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + -1.666666667a + a2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + -1.666666667a + a2 = 0 + 0.3333333333
-1.666666667a + a2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
-1.666666667a + a2 = 0.3333333333

The a term is -1.666666667a.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667a + 0.6944444447 + a2 = 0.3333333333 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667a + a2 = 0.3333333333 + 0.6944444447

Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778
0.6944444447 + -1.666666667a + a2 = 1.027777778

Factor a perfect square on the left side:
(a + -0.8333333335)(a + -0.8333333335) = 1.027777778

Calculate the square root of the right side: 1.013793755

Break this problem into two subproblems by setting 
(a + -0.8333333335) equal to 1.013793755 and -1.013793755.

Subproblem 1

a + -0.8333333335 = 1.013793755 Simplifying a + -0.8333333335 = 1.013793755 Reorder the terms: -0.8333333335 + a = 1.013793755 Solving -0.8333333335 + a = 1.013793755 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + a = 1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + a = 1.013793755 + 0.8333333335 a = 1.013793755 + 0.8333333335 Combine like terms: 1.013793755 + 0.8333333335 = 1.8471270885 a = 1.8471270885 Simplifying a = 1.8471270885

Subproblem 2

a + -0.8333333335 = -1.013793755 Simplifying a + -0.8333333335 = -1.013793755 Reorder the terms: -0.8333333335 + a = -1.013793755 Solving -0.8333333335 + a = -1.013793755 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + a = -1.013793755 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + a = -1.013793755 + 0.8333333335 a = -1.013793755 + 0.8333333335 Combine like terms: -1.013793755 + 0.8333333335 = -0.1804604215 a = -0.1804604215 Simplifying a = -0.1804604215

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.8471270885, -0.1804604215}

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