7a+29=a^2+6a+7

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Solution for 7a+29=a^2+6a+7 equation:


Simplifying
7a + 29 = a2 + 6a + 7

Reorder the terms:
29 + 7a = a2 + 6a + 7

Reorder the terms:
29 + 7a = 7 + 6a + a2

Solving
29 + 7a = 7 + 6a + a2

Solving for variable 'a'.

Reorder the terms:
29 + -7 + 7a + -6a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2

Combine like terms: 29 + -7 = 22
22 + 7a + -6a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2

Combine like terms: 7a + -6a = 1a
22 + 1a + -1a2 = 7 + 6a + a2 + -7 + -6a + -1a2

Reorder the terms:
22 + 1a + -1a2 = 7 + -7 + 6a + -6a + a2 + -1a2

Combine like terms: 7 + -7 = 0
22 + 1a + -1a2 = 0 + 6a + -6a + a2 + -1a2
22 + 1a + -1a2 = 6a + -6a + a2 + -1a2

Combine like terms: 6a + -6a = 0
22 + 1a + -1a2 = 0 + a2 + -1a2
22 + 1a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
22 + 1a + -1a2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-22 + -1a + a2 = 0

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + -1a + 22 + a2 = 0 + 22

Reorder the terms:
-22 + 22 + -1a + a2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + -1a + a2 = 0 + 22
-1a + a2 = 0 + 22

Combine like terms: 0 + 22 = 22
-1a + a2 = 22

The a term is -1a.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1a + 0.25 + a2 = 22 + 0.25

Reorder the terms:
0.25 + -1a + a2 = 22 + 0.25

Combine like terms: 22 + 0.25 = 22.25
0.25 + -1a + a2 = 22.25

Factor a perfect square on the left side:
(a + -0.5)(a + -0.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(a + -0.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

a + -0.5 = 4.716990566 Simplifying a + -0.5 = 4.716990566 Reorder the terms: -0.5 + a = 4.716990566 Solving -0.5 + a = 4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = 4.716990566 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = 4.716990566 + 0.5 a = 4.716990566 + 0.5 Combine like terms: 4.716990566 + 0.5 = 5.216990566 a = 5.216990566 Simplifying a = 5.216990566

Subproblem 2

a + -0.5 = -4.716990566 Simplifying a + -0.5 = -4.716990566 Reorder the terms: -0.5 + a = -4.716990566 Solving -0.5 + a = -4.716990566 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + a = -4.716990566 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + a = -4.716990566 + 0.5 a = -4.716990566 + 0.5 Combine like terms: -4.716990566 + 0.5 = -4.216990566 a = -4.216990566 Simplifying a = -4.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. a = {5.216990566, -4.216990566}

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