3x^2=11x+15

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Solution for 3x^2=11x+15 equation:


Simplifying
3x2 = 11x + 15

Reorder the terms:
3x2 = 15 + 11x

Solving
3x2 = 15 + 11x

Solving for variable 'x'.

Reorder the terms:
-15 + -11x + 3x2 = 15 + 11x + -15 + -11x

Reorder the terms:
-15 + -11x + 3x2 = 15 + -15 + 11x + -11x

Combine like terms: 15 + -15 = 0
-15 + -11x + 3x2 = 0 + 11x + -11x
-15 + -11x + 3x2 = 11x + -11x

Combine like terms: 11x + -11x = 0
-15 + -11x + 3x2 = 0

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

Divide each side by '3'.
-5 + -3.666666667x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -3.666666667x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + -3.666666667x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -3.666666667x + x2 = 0 + 5
-3.666666667x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
-3.666666667x + x2 = 5

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 = 5 + 3.361111114

Reorder the terms:
3.361111114 + -3.666666667x + x2 = 5 + 3.361111114

Combine like terms: 5 + 3.361111114 = 8.361111114
3.361111114 + -3.666666667x + x2 = 8.361111114

Factor a perfect square on the left side:
(x + -1.833333334)(x + -1.833333334) = 8.361111114

Calculate the square root of the right side: 2.891558596

Break this problem into two subproblems by setting 
(x + -1.833333334) equal to 2.891558596 and -2.891558596.

Subproblem 1

x + -1.833333334 = 2.891558596 Simplifying x + -1.833333334 = 2.891558596 Reorder the terms: -1.833333334 + x = 2.891558596 Solving -1.833333334 + x = 2.891558596 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 = 2.891558596 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = 2.891558596 + 1.833333334 x = 2.891558596 + 1.833333334 Combine like terms: 2.891558596 + 1.833333334 = 4.72489193 x = 4.72489193 Simplifying x = 4.72489193

Subproblem 2

x + -1.833333334 = -2.891558596 Simplifying x + -1.833333334 = -2.891558596 Reorder the terms: -1.833333334 + x = -2.891558596 Solving -1.833333334 + x = -2.891558596 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 = -2.891558596 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + x = -2.891558596 + 1.833333334 x = -2.891558596 + 1.833333334 Combine like terms: -2.891558596 + 1.833333334 = -1.058225262 x = -1.058225262 Simplifying x = -1.058225262

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.72489193, -1.058225262}

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