4A+5=A^2+3A

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Solution for 4A+5=A^2+3A equation:


Simplifying
4A + 5 = A2 + 3A

Reorder the terms:
5 + 4A = A2 + 3A

Reorder the terms:
5 + 4A = 3A + A2

Solving
5 + 4A = 3A + A2

Solving for variable 'A'.

Combine like terms: 4A + -3A = 1A
5 + 1A + -1A2 = 3A + A2 + -3A + -1A2

Reorder the terms:
5 + 1A + -1A2 = 3A + -3A + A2 + -1A2

Combine like terms: 3A + -3A = 0
5 + 1A + -1A2 = 0 + A2 + -1A2
5 + 1A + -1A2 = A2 + -1A2

Combine like terms: A2 + -1A2 = 0
5 + 1A + -1A2 = 0

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

Divide each side by '-1'.
-5 + -1A + A2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1A + 5 + A2 = 0 + 5

Reorder the terms:
-5 + 5 + -1A + A2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -1A + A2 = 0 + 5
-1A + A2 = 0 + 5

Combine like terms: 0 + 5 = 5
-1A + A2 = 5

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 = 5 + 0.25

Reorder the terms:
0.25 + -1A + A2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1A + A2 = 5.25

Factor a perfect square on the left side:
(A + -0.5)(A + -0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(A + -0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

A + -0.5 = 2.291287847 Simplifying A + -0.5 = 2.291287847 Reorder the terms: -0.5 + A = 2.291287847 Solving -0.5 + A = 2.291287847 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 = 2.291287847 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + A = 2.291287847 + 0.5 A = 2.291287847 + 0.5 Combine like terms: 2.291287847 + 0.5 = 2.791287847 A = 2.791287847 Simplifying A = 2.791287847

Subproblem 2

A + -0.5 = -2.291287847 Simplifying A + -0.5 = -2.291287847 Reorder the terms: -0.5 + A = -2.291287847 Solving -0.5 + A = -2.291287847 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 = -2.291287847 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + A = -2.291287847 + 0.5 A = -2.291287847 + 0.5 Combine like terms: -2.291287847 + 0.5 = -1.791287847 A = -1.791287847 Simplifying A = -1.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. A = {2.791287847, -1.791287847}

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