0=3e^2-4e-21

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Solution for 0=3e^2-4e-21 equation:


Simplifying
0 = 3e2 + -4e + -21

Reorder the terms:
0 = -21 + -4e + 3e2

Solving
0 = -21 + -4e + 3e2

Solving for variable 'e'.

Combine like terms: 0 + 21 = 21
21 + 4e + -3e2 = -21 + -4e + 3e2 + 21 + 4e + -3e2

Reorder the terms:
21 + 4e + -3e2 = -21 + 21 + -4e + 4e + 3e2 + -3e2

Combine like terms: -21 + 21 = 0
21 + 4e + -3e2 = 0 + -4e + 4e + 3e2 + -3e2
21 + 4e + -3e2 = -4e + 4e + 3e2 + -3e2

Combine like terms: -4e + 4e = 0
21 + 4e + -3e2 = 0 + 3e2 + -3e2
21 + 4e + -3e2 = 3e2 + -3e2

Combine like terms: 3e2 + -3e2 = 0
21 + 4e + -3e2 = 0

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

Divide each side by '-3'.
-7 + -1.333333333e + e2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -1.333333333e + 7 + e2 = 0 + 7

Reorder the terms:
-7 + 7 + -1.333333333e + e2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -1.333333333e + e2 = 0 + 7
-1.333333333e + e2 = 0 + 7

Combine like terms: 0 + 7 = 7
-1.333333333e + e2 = 7

The e term is -1.333333333e.  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.
-1.333333333e + 0.4444444442 + e2 = 7 + 0.4444444442

Reorder the terms:
0.4444444442 + -1.333333333e + e2 = 7 + 0.4444444442

Combine like terms: 7 + 0.4444444442 = 7.4444444442
0.4444444442 + -1.333333333e + e2 = 7.4444444442

Factor a perfect square on the left side:
(e + -0.6666666665)(e + -0.6666666665) = 7.4444444442

Calculate the square root of the right side: 2.728450924

Break this problem into two subproblems by setting 
(e + -0.6666666665) equal to 2.728450924 and -2.728450924.

Subproblem 1

e + -0.6666666665 = 2.728450924 Simplifying e + -0.6666666665 = 2.728450924 Reorder the terms: -0.6666666665 + e = 2.728450924 Solving -0.6666666665 + e = 2.728450924 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + e = 2.728450924 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + e = 2.728450924 + 0.6666666665 e = 2.728450924 + 0.6666666665 Combine like terms: 2.728450924 + 0.6666666665 = 3.3951175905 e = 3.3951175905 Simplifying e = 3.3951175905

Subproblem 2

e + -0.6666666665 = -2.728450924 Simplifying e + -0.6666666665 = -2.728450924 Reorder the terms: -0.6666666665 + e = -2.728450924 Solving -0.6666666665 + e = -2.728450924 Solving for variable 'e'. Move all terms containing e to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + e = -2.728450924 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + e = -2.728450924 + 0.6666666665 e = -2.728450924 + 0.6666666665 Combine like terms: -2.728450924 + 0.6666666665 = -2.0617842575 e = -2.0617842575 Simplifying e = -2.0617842575

Solution

The solution to the problem is based on the solutions from the subproblems. e = {3.3951175905, -2.0617842575}

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