-7a(a+6)+2(5+5a)=-53

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Solution for -7a(a+6)+2(5+5a)=-53 equation:


Simplifying
-7a(a + 6) + 2(5 + 5a) = -53

Reorder the terms:
-7a(6 + a) + 2(5 + 5a) = -53
(6 * -7a + a * -7a) + 2(5 + 5a) = -53
(-42a + -7a2) + 2(5 + 5a) = -53
-42a + -7a2 + (5 * 2 + 5a * 2) = -53
-42a + -7a2 + (10 + 10a) = -53

Reorder the terms:
10 + -42a + 10a + -7a2 = -53

Combine like terms: -42a + 10a = -32a
10 + -32a + -7a2 = -53

Solving
10 + -32a + -7a2 = -53

Solving for variable 'a'.

Reorder the terms:
10 + 53 + -32a + -7a2 = -53 + 53

Combine like terms: 10 + 53 = 63
63 + -32a + -7a2 = -53 + 53

Combine like terms: -53 + 53 = 0
63 + -32a + -7a2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
-9 + 4.571428571a + a2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 4.571428571a + 9 + a2 = 0 + 9

Reorder the terms:
-9 + 9 + 4.571428571a + a2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 4.571428571a + a2 = 0 + 9
4.571428571a + a2 = 0 + 9

Combine like terms: 0 + 9 = 9
4.571428571a + a2 = 9

The a term is 4.571428571a.  Take half its coefficient (2.285714286).
Square it (5.224489797) and add it to both sides.

Add '5.224489797' to each side of the equation.
4.571428571a + 5.224489797 + a2 = 9 + 5.224489797

Reorder the terms:
5.224489797 + 4.571428571a + a2 = 9 + 5.224489797

Combine like terms: 9 + 5.224489797 = 14.224489797
5.224489797 + 4.571428571a + a2 = 14.224489797

Factor a perfect square on the left side:
(a + 2.285714286)(a + 2.285714286) = 14.224489797

Calculate the square root of the right side: 3.771536795

Break this problem into two subproblems by setting 
(a + 2.285714286) equal to 3.771536795 and -3.771536795.

Subproblem 1

a + 2.285714286 = 3.771536795 Simplifying a + 2.285714286 = 3.771536795 Reorder the terms: 2.285714286 + a = 3.771536795 Solving 2.285714286 + a = 3.771536795 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = 3.771536795 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = 3.771536795 + -2.285714286 a = 3.771536795 + -2.285714286 Combine like terms: 3.771536795 + -2.285714286 = 1.485822509 a = 1.485822509 Simplifying a = 1.485822509

Subproblem 2

a + 2.285714286 = -3.771536795 Simplifying a + 2.285714286 = -3.771536795 Reorder the terms: 2.285714286 + a = -3.771536795 Solving 2.285714286 + a = -3.771536795 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = -3.771536795 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = -3.771536795 + -2.285714286 a = -3.771536795 + -2.285714286 Combine like terms: -3.771536795 + -2.285714286 = -6.057251081 a = -6.057251081 Simplifying a = -6.057251081

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.485822509, -6.057251081}

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