15a^2-31=3-7a

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Solution for 15a^2-31=3-7a equation:


Simplifying
15a2 + -31 = 3 + -7a

Reorder the terms:
-31 + 15a2 = 3 + -7a

Solving
-31 + 15a2 = 3 + -7a

Solving for variable 'a'.

Reorder the terms:
-31 + -3 + 7a + 15a2 = 3 + -7a + -3 + 7a

Combine like terms: -31 + -3 = -34
-34 + 7a + 15a2 = 3 + -7a + -3 + 7a

Reorder the terms:
-34 + 7a + 15a2 = 3 + -3 + -7a + 7a

Combine like terms: 3 + -3 = 0
-34 + 7a + 15a2 = 0 + -7a + 7a
-34 + 7a + 15a2 = -7a + 7a

Combine like terms: -7a + 7a = 0
-34 + 7a + 15a2 = 0

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

Divide each side by '15'.
-2.266666667 + 0.4666666667a + a2 = 0

Move the constant term to the right:

Add '2.266666667' to each side of the equation.
-2.266666667 + 0.4666666667a + 2.266666667 + a2 = 0 + 2.266666667

Reorder the terms:
-2.266666667 + 2.266666667 + 0.4666666667a + a2 = 0 + 2.266666667

Combine like terms: -2.266666667 + 2.266666667 = 0.000000000
0.000000000 + 0.4666666667a + a2 = 0 + 2.266666667
0.4666666667a + a2 = 0 + 2.266666667

Combine like terms: 0 + 2.266666667 = 2.266666667
0.4666666667a + a2 = 2.266666667

The a term is 0.4666666667a.  Take half its coefficient (0.2333333334).
Square it (0.05444444448) and add it to both sides.

Add '0.05444444448' to each side of the equation.
0.4666666667a + 0.05444444448 + a2 = 2.266666667 + 0.05444444448

Reorder the terms:
0.05444444448 + 0.4666666667a + a2 = 2.266666667 + 0.05444444448

Combine like terms: 2.266666667 + 0.05444444448 = 2.32111111148
0.05444444448 + 0.4666666667a + a2 = 2.32111111148

Factor a perfect square on the left side:
(a + 0.2333333334)(a + 0.2333333334) = 2.32111111148

Calculate the square root of the right side: 1.523519318

Break this problem into two subproblems by setting 
(a + 0.2333333334) equal to 1.523519318 and -1.523519318.

Subproblem 1

a + 0.2333333334 = 1.523519318 Simplifying a + 0.2333333334 = 1.523519318 Reorder the terms: 0.2333333334 + a = 1.523519318 Solving 0.2333333334 + a = 1.523519318 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2333333334' to each side of the equation. 0.2333333334 + -0.2333333334 + a = 1.523519318 + -0.2333333334 Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000 0.0000000000 + a = 1.523519318 + -0.2333333334 a = 1.523519318 + -0.2333333334 Combine like terms: 1.523519318 + -0.2333333334 = 1.2901859846 a = 1.2901859846 Simplifying a = 1.2901859846

Subproblem 2

a + 0.2333333334 = -1.523519318 Simplifying a + 0.2333333334 = -1.523519318 Reorder the terms: 0.2333333334 + a = -1.523519318 Solving 0.2333333334 + a = -1.523519318 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.2333333334' to each side of the equation. 0.2333333334 + -0.2333333334 + a = -1.523519318 + -0.2333333334 Combine like terms: 0.2333333334 + -0.2333333334 = 0.0000000000 0.0000000000 + a = -1.523519318 + -0.2333333334 a = -1.523519318 + -0.2333333334 Combine like terms: -1.523519318 + -0.2333333334 = -1.7568526514 a = -1.7568526514 Simplifying a = -1.7568526514

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.2901859846, -1.7568526514}

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