So, the thought came to me of solving a cubic equation. I've done it a few times before, but it has been years. I know how to solve quadratics, so I was thinking that if I could reduce the cubic to a quadratic, then I could solve the quadratic for [imath]x[/imath] and thus get my cubic solutions. Everything went smoothly until I reached a point where I have an equation of the form [imath]ax^2 + bx + c + \frac{d}{x} = 0[/imath]. This form is the quadratic reduction of the original cubic of the form [imath]ax^3 + bx^2 + cx + d = 0[/imath]. I'm wondering how I should factor this, so that I can then solve for [imath]x[/imath]. Here's what I have so far:
Solve for x
[math]x^3 - 6x^2 + 2x + 3 = 0[/math]
Subtract 3 from both sides to get
[math]x^3 - 6x^2 + 2x = -3[/math]
Multiply both sides by [imath]\frac{1}{x}[/imath] to get
[math]x^2 - 6x + 2 = \frac{-3}{x}[/math]
Add [imath]\frac{3}{x}[/imath] to both sides to get
[math]x^2 - 6x + 2 + \frac{3}{x} = 0[/math]
And here I am, stuck with this [imath]\frac{d}{x}[/imath] term in my quadratic and not knowing how to factor a quadratic with such a term in it. When I try to find an answer to my question, all I get is how to factor a quadratic that has fractional coefficients, which is not at all the same as factoring a quadratic with a [imath]\frac{d}{x}[/imath] term. So, how can I go about factoring this quadratic with a [imath]\frac{d}{x}[/imath] term in it? Do I have to move the [imath]\frac{d}{x}[/imath] term back to the other side and then factor [imath]x^2 - 6x + 2[/imath] as if it equals 0 and then solve for [imath]x[/imath]? Or is there another way to go about it that keeps the right side equal to 0?
Solve for x
[math]x^3 - 6x^2 + 2x + 3 = 0[/math]
Subtract 3 from both sides to get
[math]x^3 - 6x^2 + 2x = -3[/math]
Multiply both sides by [imath]\frac{1}{x}[/imath] to get
[math]x^2 - 6x + 2 = \frac{-3}{x}[/math]
Add [imath]\frac{3}{x}[/imath] to both sides to get
[math]x^2 - 6x + 2 + \frac{3}{x} = 0[/math]
And here I am, stuck with this [imath]\frac{d}{x}[/imath] term in my quadratic and not knowing how to factor a quadratic with such a term in it. When I try to find an answer to my question, all I get is how to factor a quadratic that has fractional coefficients, which is not at all the same as factoring a quadratic with a [imath]\frac{d}{x}[/imath] term. So, how can I go about factoring this quadratic with a [imath]\frac{d}{x}[/imath] term in it? Do I have to move the [imath]\frac{d}{x}[/imath] term back to the other side and then factor [imath]x^2 - 6x + 2[/imath] as if it equals 0 and then solve for [imath]x[/imath]? Or is there another way to go about it that keeps the right side equal to 0?