3x^2=5x+11

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Solution for 3x^2=5x+11 equation:


Simplifying
3x2 = 5x + 11

Reorder the terms:
3x2 = 11 + 5x

Solving
3x2 = 11 + 5x

Solving for variable 'x'.

Reorder the terms:
-11 + -5x + 3x2 = 11 + 5x + -11 + -5x

Reorder the terms:
-11 + -5x + 3x2 = 11 + -11 + 5x + -5x

Combine like terms: 11 + -11 = 0
-11 + -5x + 3x2 = 0 + 5x + -5x
-11 + -5x + 3x2 = 5x + -5x

Combine like terms: 5x + -5x = 0
-11 + -5x + 3x2 = 0

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

Divide each side by '3'.
-3.666666667 + -1.666666667x + x2 = 0

Move the constant term to the right:

Add '3.666666667' to each side of the equation.
-3.666666667 + -1.666666667x + 3.666666667 + x2 = 0 + 3.666666667

Reorder the terms:
-3.666666667 + 3.666666667 + -1.666666667x + x2 = 0 + 3.666666667

Combine like terms: -3.666666667 + 3.666666667 = 0.000000000
0.000000000 + -1.666666667x + x2 = 0 + 3.666666667
-1.666666667x + x2 = 0 + 3.666666667

Combine like terms: 0 + 3.666666667 = 3.666666667
-1.666666667x + x2 = 3.666666667

The x term is -1.666666667x.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667x + 0.6944444447 + x2 = 3.666666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667x + x2 = 3.666666667 + 0.6944444447

Combine like terms: 3.666666667 + 0.6944444447 = 4.3611111117
0.6944444447 + -1.666666667x + x2 = 4.3611111117

Factor a perfect square on the left side:
(x + -0.8333333335)(x + -0.8333333335) = 4.3611111117

Calculate the square root of the right side: 2.088327348

Break this problem into two subproblems by setting 
(x + -0.8333333335) equal to 2.088327348 and -2.088327348.

Subproblem 1

x + -0.8333333335 = 2.088327348 Simplifying x + -0.8333333335 = 2.088327348 Reorder the terms: -0.8333333335 + x = 2.088327348 Solving -0.8333333335 + x = 2.088327348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = 2.088327348 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = 2.088327348 + 0.8333333335 x = 2.088327348 + 0.8333333335 Combine like terms: 2.088327348 + 0.8333333335 = 2.9216606815 x = 2.9216606815 Simplifying x = 2.9216606815

Subproblem 2

x + -0.8333333335 = -2.088327348 Simplifying x + -0.8333333335 = -2.088327348 Reorder the terms: -0.8333333335 + x = -2.088327348 Solving -0.8333333335 + x = -2.088327348 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + x = -2.088327348 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + x = -2.088327348 + 0.8333333335 x = -2.088327348 + 0.8333333335 Combine like terms: -2.088327348 + 0.8333333335 = -1.2549940145 x = -1.2549940145 Simplifying x = -1.2549940145

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.9216606815, -1.2549940145}

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