How are cubic equations converted into this form?

  • Thread starter autodidude
  • Start date
  • Tags
    Cubic Form
In summary, the formula y = a(x-b)^3 + c involves completing the cube, similar to completing the square for quadratic equations. However, not all cubic equations can be converted into this form and the coefficients of x^3, x^2, and x must be related correctly for it to work. This is why there is no standard name for this operation.
  • #1
autodidude
333
0
y = a(x-b)^3 + c

I'm not sure what it's called, my book doesn't mention on how it's derived.
 
Mathematics news on Phys.org
  • #2
Well, not all cubic equations can be converted into this form. For example, x3+4x2+3x+1 can only be converted into (x+1)3+x2, and unless you let c be non-constant, it can't be put into the form you're looking for.
 
  • #3
I guess it would be called "completing the cube" like "completing the square".

But to be able to "complete the square" you must be given coefficients of [itex]x^2[/itex] and [itex]x[/itex] while to "complete the cube" you would have to be given coefficients of [itex]x^3[/itex], [itex]x^2[/itex], and x. And, as Char.Limit said, unlike completing the square, if those coefficients are not related correctly, it will be impossible. Which is probably why there is not standard name for the operation!
 

What is the standard form of a cubic equation?

The standard form of a cubic equation is ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and a is not equal to 0.

How do you convert a cubic equation into standard form?

To convert a cubic equation into standard form, you must first ensure that the equation is in descending order. Then, you can factor out the leading coefficient and use the quadratic formula to solve for the remaining roots.

Why is it important to convert a cubic equation into standard form?

Converting a cubic equation into standard form allows for easier identification of the leading coefficient and the constant term, which can provide insights into the behavior of the equation. It also simplifies the process of solving the equation.

Can all cubic equations be converted into standard form?

Yes, all cubic equations can be converted into standard form as long as the equation does not contain any imaginary or complex numbers.

What other forms can a cubic equation be written in?

Besides standard form, a cubic equation can also be written in factored form, which is (x - r)(x - s)(x - t) = 0, where r, s, and t are the roots of the equation. It can also be written in vertex form, which is a(x - h)^3 + k = 0, where (h, k) is the vertex of the cubic function.

Similar threads

  • General Math
Replies
16
Views
3K
Replies
9
Views
1K
Replies
9
Views
1K
  • General Math
Replies
1
Views
604
  • General Math
Replies
2
Views
1K
  • General Math
Replies
3
Views
1K
  • General Math
Replies
2
Views
1K
Replies
8
Views
3K
  • General Math
Replies
10
Views
2K
Replies
1
Views
689
Back
Top