15a^2-3a=3+7a

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Solution for 15a^2-3a=3+7a equation:


Simplifying
15a2 + -3a = 3 + 7a

Reorder the terms:
-3a + 15a2 = 3 + 7a

Solving
-3a + 15a2 = 3 + 7a

Solving for variable 'a'.

Reorder the terms:
-3 + -3a + -7a + 15a2 = 3 + 7a + -3 + -7a

Combine like terms: -3a + -7a = -10a
-3 + -10a + 15a2 = 3 + 7a + -3 + -7a

Reorder the terms:
-3 + -10a + 15a2 = 3 + -3 + 7a + -7a

Combine like terms: 3 + -3 = 0
-3 + -10a + 15a2 = 0 + 7a + -7a
-3 + -10a + 15a2 = 7a + -7a

Combine like terms: 7a + -7a = 0
-3 + -10a + 15a2 = 0

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

Divide each side by '15'.
-0.2 + -0.6666666667a + a2 = 0

Move the constant term to the right:

Add '0.2' to each side of the equation.
-0.2 + -0.6666666667a + 0.2 + a2 = 0 + 0.2

Reorder the terms:
-0.2 + 0.2 + -0.6666666667a + a2 = 0 + 0.2

Combine like terms: -0.2 + 0.2 = 0.0
0.0 + -0.6666666667a + a2 = 0 + 0.2
-0.6666666667a + a2 = 0 + 0.2

Combine like terms: 0 + 0.2 = 0.2
-0.6666666667a + a2 = 0.2

The a term is -0.6666666667a.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667a + 0.1111111112 + a2 = 0.2 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667a + a2 = 0.2 + 0.1111111112

Combine like terms: 0.2 + 0.1111111112 = 0.3111111112
0.1111111112 + -0.6666666667a + a2 = 0.3111111112

Factor a perfect square on the left side:
(a + -0.3333333334)(a + -0.3333333334) = 0.3111111112

Calculate the square root of the right side: 0.557773351

Break this problem into two subproblems by setting 
(a + -0.3333333334) equal to 0.557773351 and -0.557773351.

Subproblem 1

a + -0.3333333334 = 0.557773351 Simplifying a + -0.3333333334 = 0.557773351 Reorder the terms: -0.3333333334 + a = 0.557773351 Solving -0.3333333334 + a = 0.557773351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = 0.557773351 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = 0.557773351 + 0.3333333334 a = 0.557773351 + 0.3333333334 Combine like terms: 0.557773351 + 0.3333333334 = 0.8911066844 a = 0.8911066844 Simplifying a = 0.8911066844

Subproblem 2

a + -0.3333333334 = -0.557773351 Simplifying a + -0.3333333334 = -0.557773351 Reorder the terms: -0.3333333334 + a = -0.557773351 Solving -0.3333333334 + a = -0.557773351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + a = -0.557773351 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + a = -0.557773351 + 0.3333333334 a = -0.557773351 + 0.3333333334 Combine like terms: -0.557773351 + 0.3333333334 = -0.2244400176 a = -0.2244400176 Simplifying a = -0.2244400176

Solution

The solution to the problem is based on the solutions from the subproblems. a = {0.8911066844, -0.2244400176}

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