3a^2=4a+3

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Solution for 3a^2=4a+3 equation:


Simplifying
3a2 = 4a + 3

Reorder the terms:
3a2 = 3 + 4a

Solving
3a2 = 3 + 4a

Solving for variable 'a'.

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

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

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

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

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

Divide each side by '3'.
-1 + -1.333333333a + a2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1.333333333a + 1 + a2 = 0 + 1

Reorder the terms:
-1 + 1 + -1.333333333a + a2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1.333333333a + a2 = 0 + 1
-1.333333333a + a2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1.333333333a + a2 = 1

The a term is -1.333333333a.  Take half its coefficient (-0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
-1.333333333a + 0.4444444442 + a2 = 1 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333a + a2 = 1 + 0.4444444442

Combine like terms: 1 + 0.4444444442 = 1.4444444442
0.4444444442 + -1.333333333a + a2 = 1.4444444442

Factor a perfect square on the left side:
(a + -0.6666666665)(a + -0.6666666665) = 1.4444444442

Calculate the square root of the right side: 1.201850425

Break this problem into two subproblems by setting 
(a + -0.6666666665) equal to 1.201850425 and -1.201850425.

Subproblem 1

a + -0.6666666665 = 1.201850425 Simplifying a + -0.6666666665 = 1.201850425 Reorder the terms: -0.6666666665 + a = 1.201850425 Solving -0.6666666665 + a = 1.201850425 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = 1.201850425 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = 1.201850425 + 0.6666666665 a = 1.201850425 + 0.6666666665 Combine like terms: 1.201850425 + 0.6666666665 = 1.8685170915 a = 1.8685170915 Simplifying a = 1.8685170915

Subproblem 2

a + -0.6666666665 = -1.201850425 Simplifying a + -0.6666666665 = -1.201850425 Reorder the terms: -0.6666666665 + a = -1.201850425 Solving -0.6666666665 + a = -1.201850425 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + a = -1.201850425 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + a = -1.201850425 + 0.6666666665 a = -1.201850425 + 0.6666666665 Combine like terms: -1.201850425 + 0.6666666665 = -0.5351837585 a = -0.5351837585 Simplifying a = -0.5351837585

Solution

The solution to the problem is based on the solutions from the subproblems. a = {1.8685170915, -0.5351837585}

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