2a(2a+7)-4+2=10

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Solution for 2a(2a+7)-4+2=10 equation: 


Simplifying
2a(2a + 7) + -4 + 2 = 10

Reorder the terms:
2a(7 + 2a) + -4 + 2 = 10
(7 * 2a + 2a * 2a) + -4 + 2 = 10
(14a + 4a2) + -4 + 2 = 10

Reorder the terms:
-4 + 2 + 14a + 4a2 = 10

Combine like terms: -4 + 2 = -2
-2 + 14a + 4a2 = 10

Solving
-2 + 14a + 4a2 = 10

Solving for variable 'a'.

Reorder the terms:
-2 + -10 + 14a + 4a2 = 10 + -10

Combine like terms: -2 + -10 = -12
-12 + 14a + 4a2 = 10 + -10

Combine like terms: 10 + -10 = 0
-12 + 14a + 4a2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-6 + 7a + 2a2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-6 + 7a + 2a2)' equal to zero and attempt to solve:

Simplifying
-6 + 7a + 2a2 = 0

Solving
-6 + 7a + 2a2 = 0

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

Divide each side by '2'.
-3 + 3.5a + a2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + 3.5a + 3 + a2 = 0 + 3

Reorder the terms:
-3 + 3 + 3.5a + a2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + 3.5a + a2 = 0 + 3
3.5a + a2 = 0 + 3

Combine like terms: 0 + 3 = 3
3.5a + a2 = 3

The a term is 3.5a.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5a + 3.0625 + a2 = 3 + 3.0625

Reorder the terms:
3.0625 + 3.5a + a2 = 3 + 3.0625

Combine like terms: 3 + 3.0625 = 6.0625
3.0625 + 3.5a + a2 = 6.0625

Factor a perfect square on the left side:
(a + 1.75)(a + 1.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(a + 1.75) equal to 2.46221445 and -2.46221445.


Subproblem 1
a + 1.75 = 2.46221445

Simplifying
a + 1.75 = 2.46221445

Reorder the terms:
1.75 + a = 2.46221445

Solving
1.75 + a = 2.46221445

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + a = 2.46221445 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + a = 2.46221445 + -1.75
a = 2.46221445 + -1.75

Combine like terms: 2.46221445 + -1.75 = 0.71221445
a = 0.71221445

Simplifying
a = 0.71221445


Subproblem 2
a + 1.75 = -2.46221445

Simplifying
a + 1.75 = -2.46221445

Reorder the terms:
1.75 + a = -2.46221445

Solving
1.75 + a = -2.46221445

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Add '-1.75' to each side of the equation.
1.75 + -1.75 + a = -2.46221445 + -1.75

Combine like terms: 1.75 + -1.75 = 0.00
0.00 + a = -2.46221445 + -1.75
a = -2.46221445 + -1.75

Combine like terms: -2.46221445 + -1.75 = -4.21221445
a = -4.21221445

Simplifying
a = -4.21221445


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.71221445, -4.21221445}
Solution
a = {0.71221445, -4.21221445}

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