2x+3x^2=90

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Solution for 2x+3x^2=90 equation:


Simplifying
2x + 3x2 = 90

Solving
2x + 3x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 2x + 3x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 2x + 3x2 = 0

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

Divide each side by '3'.
-30 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + 0.6666666667x + 30 + x2 = 0 + 30

Reorder the terms:
-30 + 30 + 0.6666666667x + x2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + 0.6666666667x + x2 = 0 + 30
0.6666666667x + x2 = 0 + 30

Combine like terms: 0 + 30 = 30
0.6666666667x + x2 = 30

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 30 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 30 + 0.1111111112

Combine like terms: 30 + 0.1111111112 = 30.1111111112
0.1111111112 + 0.6666666667x + x2 = 30.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 30.1111111112

Calculate the square root of the right side: 5.487359211

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 5.487359211 and -5.487359211.

Subproblem 1

x + 0.3333333334 = 5.487359211 Simplifying x + 0.3333333334 = 5.487359211 Reorder the terms: 0.3333333334 + x = 5.487359211 Solving 0.3333333334 + x = 5.487359211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 5.487359211 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 5.487359211 + -0.3333333334 x = 5.487359211 + -0.3333333334 Combine like terms: 5.487359211 + -0.3333333334 = 5.1540258776 x = 5.1540258776 Simplifying x = 5.1540258776

Subproblem 2

x + 0.3333333334 = -5.487359211 Simplifying x + 0.3333333334 = -5.487359211 Reorder the terms: 0.3333333334 + x = -5.487359211 Solving 0.3333333334 + x = -5.487359211 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -5.487359211 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -5.487359211 + -0.3333333334 x = -5.487359211 + -0.3333333334 Combine like terms: -5.487359211 + -0.3333333334 = -5.8206925444 x = -5.8206925444 Simplifying x = -5.8206925444

Solution

The solution to the problem is based on the solutions from the subproblems. x = {5.1540258776, -5.8206925444}

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