3x^2+11x-45=0

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Solution for 3x^2+11x-45=0 equation:


Simplifying
3x2 + 11x + -45 = 0

Reorder the terms:
-45 + 11x + 3x2 = 0

Solving
-45 + 11x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
-15 + 3.666666667x + x2 = 0

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + 3.666666667x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + 3.666666667x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + 3.666666667x + x2 = 0 + 15
3.666666667x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
3.666666667x + x2 = 15

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 = 15 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667x + x2 = 15 + 3.361111114

Combine like terms: 15 + 3.361111114 = 18.361111114
3.361111114 + 3.666666667x + x2 = 18.361111114

Factor a perfect square on the left side:
(x + 1.833333334)(x + 1.833333334) = 18.361111114

Calculate the square root of the right side: 4.284986711

Break this problem into two subproblems by setting 
(x + 1.833333334) equal to 4.284986711 and -4.284986711.

Subproblem 1

x + 1.833333334 = 4.284986711 Simplifying x + 1.833333334 = 4.284986711 Reorder the terms: 1.833333334 + x = 4.284986711 Solving 1.833333334 + x = 4.284986711 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 = 4.284986711 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = 4.284986711 + -1.833333334 x = 4.284986711 + -1.833333334 Combine like terms: 4.284986711 + -1.833333334 = 2.451653377 x = 2.451653377 Simplifying x = 2.451653377

Subproblem 2

x + 1.833333334 = -4.284986711 Simplifying x + 1.833333334 = -4.284986711 Reorder the terms: 1.833333334 + x = -4.284986711 Solving 1.833333334 + x = -4.284986711 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 = -4.284986711 + -1.833333334 Combine like terms: 1.833333334 + -1.833333334 = 0.000000000 0.000000000 + x = -4.284986711 + -1.833333334 x = -4.284986711 + -1.833333334 Combine like terms: -4.284986711 + -1.833333334 = -6.118320045 x = -6.118320045 Simplifying x = -6.118320045

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.451653377, -6.118320045}

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