14=3a+a^2

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Solution for 14=3a+a^2 equation:


Simplifying
14 = 3a + a2

Solving
14 = 3a + a2

Solving for variable 'a'.

Reorder the terms:
14 + -3a + -1a2 = 3a + -3a + a2 + -1a2

Combine like terms: 3a + -3a = 0
14 + -3a + -1a2 = 0 + a2 + -1a2
14 + -3a + -1a2 = a2 + -1a2

Combine like terms: a2 + -1a2 = 0
14 + -3a + -1a2 = 0

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

Divide each side by '-1'.
-14 + 3a + a2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 3a + 14 + a2 = 0 + 14

Reorder the terms:
-14 + 14 + 3a + a2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 3a + a2 = 0 + 14
3a + a2 = 0 + 14

Combine like terms: 0 + 14 = 14
3a + a2 = 14

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = 14 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 14 + 2.25

Combine like terms: 14 + 2.25 = 16.25
2.25 + 3a + a2 = 16.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 16.25

Calculate the square root of the right side: 4.031128874

Break this problem into two subproblems by setting 
(a + 1.5) equal to 4.031128874 and -4.031128874.

Subproblem 1

a + 1.5 = 4.031128874 Simplifying a + 1.5 = 4.031128874 Reorder the terms: 1.5 + a = 4.031128874 Solving 1.5 + a = 4.031128874 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 4.031128874 + -1.5 a = 4.031128874 + -1.5 Combine like terms: 4.031128874 + -1.5 = 2.531128874 a = 2.531128874 Simplifying a = 2.531128874

Subproblem 2

a + 1.5 = -4.031128874 Simplifying a + 1.5 = -4.031128874 Reorder the terms: 1.5 + a = -4.031128874 Solving 1.5 + a = -4.031128874 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -4.031128874 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -4.031128874 + -1.5 a = -4.031128874 + -1.5 Combine like terms: -4.031128874 + -1.5 = -5.531128874 a = -5.531128874 Simplifying a = -5.531128874

Solution

The solution to the problem is based on the solutions from the subproblems. a = {2.531128874, -5.531128874}

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