4a(a)=2a+3

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Solution for 4a(a)=2a+3 equation:


Simplifying
4a(a) = 2a + 3

Multiply a * a
4a2 = 2a + 3

Reorder the terms:
4a2 = 3 + 2a

Solving
4a2 = 3 + 2a

Solving for variable 'a'.

Reorder the terms:
-3 + -2a + 4a2 = 3 + 2a + -3 + -2a

Reorder the terms:
-3 + -2a + 4a2 = 3 + -3 + 2a + -2a

Combine like terms: 3 + -3 = 0
-3 + -2a + 4a2 = 0 + 2a + -2a
-3 + -2a + 4a2 = 2a + -2a

Combine like terms: 2a + -2a = 0
-3 + -2a + 4a2 = 0

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

Divide each side by '4'.
-0.75 + -0.5a + a2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + -0.5a + 0.75 + a2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + -0.5a + a2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + -0.5a + a2 = 0 + 0.75
-0.5a + a2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
-0.5a + a2 = 0.75

The a term is -0.5a.  Take half its coefficient (-0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
-0.5a + 0.0625 + a2 = 0.75 + 0.0625

Reorder the terms:
0.0625 + -0.5a + a2 = 0.75 + 0.0625

Combine like terms: 0.75 + 0.0625 = 0.8125
0.0625 + -0.5a + a2 = 0.8125

Factor a perfect square on the left side:
(a + -0.25)(a + -0.25) = 0.8125

Calculate the square root of the right side: 0.901387819

Break this problem into two subproblems by setting 
(a + -0.25) equal to 0.901387819 and -0.901387819.

Subproblem 1

a + -0.25 = 0.901387819 Simplifying a + -0.25 = 0.901387819 Reorder the terms: -0.25 + a = 0.901387819 Solving -0.25 + a = 0.901387819 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = 0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 0.901387819 + 0.25 a = 0.901387819 + 0.25 Combine like terms: 0.901387819 + 0.25 = 1.151387819 a = 1.151387819 Simplifying a = 1.151387819

Subproblem 2

a + -0.25 = -0.901387819 Simplifying a + -0.25 = -0.901387819 Reorder the terms: -0.25 + a = -0.901387819 Solving -0.25 + a = -0.901387819 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + a = -0.901387819 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -0.901387819 + 0.25 a = -0.901387819 + 0.25 Combine like terms: -0.901387819 + 0.25 = -0.651387819 a = -0.651387819 Simplifying a = -0.651387819

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.151387819, -0.651387819}

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