a(5+3a)=22-7

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Solution for a(5+3a)=22-7 equation:


Simplifying
a(5 + 3a) = 22 + -7
(5 * a + 3a * a) = 22 + -7
(5a + 3a2) = 22 + -7

Combine like terms: 22 + -7 = 15
5a + 3a2 = 15

Solving
5a + 3a2 = 15

Solving for variable 'a'.

Reorder the terms:
-15 + 5a + 3a2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 5a + 3a2 = 0

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

Divide each side by '3'.
-5 + 1.666666667a + a2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 1.666666667a + 5 + a2 = 0 + 5

Reorder the terms:
-5 + 5 + 1.666666667a + a2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 1.666666667a + a2 = 0 + 5
1.666666667a + a2 = 0 + 5

Combine like terms: 0 + 5 = 5
1.666666667a + a2 = 5

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 = 5 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667a + a2 = 5 + 0.6944444447

Combine like terms: 5 + 0.6944444447 = 5.6944444447
0.6944444447 + 1.666666667a + a2 = 5.6944444447

Factor a perfect square on the left side:
(a + 0.8333333335)(a + 0.8333333335) = 5.6944444447

Calculate the square root of the right side: 2.386303511

Break this problem into two subproblems by setting 
(a + 0.8333333335) equal to 2.386303511 and -2.386303511.

Subproblem 1

a + 0.8333333335 = 2.386303511 Simplifying a + 0.8333333335 = 2.386303511 Reorder the terms: 0.8333333335 + a = 2.386303511 Solving 0.8333333335 + a = 2.386303511 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 = 2.386303511 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = 2.386303511 + -0.8333333335 a = 2.386303511 + -0.8333333335 Combine like terms: 2.386303511 + -0.8333333335 = 1.5529701775 a = 1.5529701775 Simplifying a = 1.5529701775

Subproblem 2

a + 0.8333333335 = -2.386303511 Simplifying a + 0.8333333335 = -2.386303511 Reorder the terms: 0.8333333335 + a = -2.386303511 Solving 0.8333333335 + a = -2.386303511 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 = -2.386303511 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = -2.386303511 + -0.8333333335 a = -2.386303511 + -0.8333333335 Combine like terms: -2.386303511 + -0.8333333335 = -3.2196368445 a = -3.2196368445 Simplifying a = -3.2196368445

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.5529701775, -3.2196368445}

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