3x^2=13x+90

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Solution for 3x^2=13x+90 equation:


Simplifying
3x2 = 13x + 90

Reorder the terms:
3x2 = 90 + 13x

Solving
3x2 = 90 + 13x

Solving for variable 'x'.

Reorder the terms:
-90 + -13x + 3x2 = 90 + 13x + -90 + -13x

Reorder the terms:
-90 + -13x + 3x2 = 90 + -90 + 13x + -13x

Combine like terms: 90 + -90 = 0
-90 + -13x + 3x2 = 0 + 13x + -13x
-90 + -13x + 3x2 = 13x + -13x

Combine like terms: 13x + -13x = 0
-90 + -13x + 3x2 = 0

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

Divide each side by '3'.
-30 + -4.333333333x + x2 = 0

Move the constant term to the right:

Add '30' to each side of the equation.
-30 + -4.333333333x + 30 + x2 = 0 + 30

Reorder the terms:
-30 + 30 + -4.333333333x + x2 = 0 + 30

Combine like terms: -30 + 30 = 0
0 + -4.333333333x + x2 = 0 + 30
-4.333333333x + x2 = 0 + 30

Combine like terms: 0 + 30 = 30
-4.333333333x + x2 = 30

The x term is -4.333333333x.  Take half its coefficient (-2.166666667).
Square it (4.694444446) and add it to both sides.

Add '4.694444446' to each side of the equation.
-4.333333333x + 4.694444446 + x2 = 30 + 4.694444446

Reorder the terms:
4.694444446 + -4.333333333x + x2 = 30 + 4.694444446

Combine like terms: 30 + 4.694444446 = 34.694444446
4.694444446 + -4.333333333x + x2 = 34.694444446

Factor a perfect square on the left side:
(x + -2.166666667)(x + -2.166666667) = 34.694444446

Calculate the square root of the right side: 5.890199016

Break this problem into two subproblems by setting 
(x + -2.166666667) equal to 5.890199016 and -5.890199016.

Subproblem 1

x + -2.166666667 = 5.890199016 Simplifying x + -2.166666667 = 5.890199016 Reorder the terms: -2.166666667 + x = 5.890199016 Solving -2.166666667 + x = 5.890199016 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + x = 5.890199016 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = 5.890199016 + 2.166666667 x = 5.890199016 + 2.166666667 Combine like terms: 5.890199016 + 2.166666667 = 8.056865683 x = 8.056865683 Simplifying x = 8.056865683

Subproblem 2

x + -2.166666667 = -5.890199016 Simplifying x + -2.166666667 = -5.890199016 Reorder the terms: -2.166666667 + x = -5.890199016 Solving -2.166666667 + x = -5.890199016 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.166666667' to each side of the equation. -2.166666667 + 2.166666667 + x = -5.890199016 + 2.166666667 Combine like terms: -2.166666667 + 2.166666667 = 0.000000000 0.000000000 + x = -5.890199016 + 2.166666667 x = -5.890199016 + 2.166666667 Combine like terms: -5.890199016 + 2.166666667 = -3.723532349 x = -3.723532349 Simplifying x = -3.723532349

Solution

The solution to the problem is based on the solutions from the subproblems. x = {8.056865683, -3.723532349}

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