3x^2=11x+21

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Solution for 3x^2=11x+21 equation:


Simplifying
3x2 = 11x + 21

Reorder the terms:
3x2 = 21 + 11x

Solving
3x2 = 21 + 11x

Solving for variable 'x'.

Reorder the terms:
-21 + -11x + 3x2 = 21 + 11x + -21 + -11x

Reorder the terms:
-21 + -11x + 3x2 = 21 + -21 + 11x + -11x

Combine like terms: 21 + -21 = 0
-21 + -11x + 3x2 = 0 + 11x + -11x
-21 + -11x + 3x2 = 11x + -11x

Combine like terms: 11x + -11x = 0
-21 + -11x + 3x2 = 0

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

Divide each side by '3'.
-7 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + -3.666666667x + 7 + x2 = 0 + 7

Reorder the terms:
-7 + 7 + -3.666666667x + x2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + -3.666666667x + x2 = 0 + 7
-3.666666667x + x2 = 0 + 7

Combine like terms: 0 + 7 = 7
-3.666666667x + x2 = 7

The x term is -3.666666667x.  Take half its coefficient (-1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
-3.666666667x + 3.361111114 + x2 = 7 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = 7 + 3.361111114

Combine like terms: 7 + 3.361111114 = 10.361111114
3.361111114 + -3.666666667x + x2 = 10.361111114

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 10.361111114

Calculate the square root of the right side: 3.218867986

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 3.218867986 and -3.218867986.

Subproblem 1

x + -1.833333334 = 3.218867986 Simplifying x + -1.833333334 = 3.218867986 Reorder the terms: -1.833333334 + x = 3.218867986 Solving -1.833333334 + x = 3.218867986 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = 3.218867986 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 3.218867986 + 1.833333334 x = 3.218867986 + 1.833333334 Combine like terms: 3.218867986 + 1.833333334 = 5.05220132 x = 5.05220132 Simplifying x = 5.05220132

Subproblem 2

x + -1.833333334 = -3.218867986 Simplifying x + -1.833333334 = -3.218867986 Reorder the terms: -1.833333334 + x = -3.218867986 Solving -1.833333334 + x = -3.218867986 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + x = -3.218867986 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -3.218867986 + 1.833333334 x = -3.218867986 + 1.833333334 Combine like terms: -3.218867986 + 1.833333334 = -1.385534652 x = -1.385534652 Simplifying x = -1.385534652

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.05220132, -1.385534652}

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