Find H

TheJason

Staff member
Moderator
There should be three answers (roots) (post edited).

$$\displaystyle 15 + 3\,h^{3} = 16 - 9\,h^{3}$$

$$\displaystyle 15 -15 + 3\,h^{3} = 16 - 15 - 9\,h^{3}$$

$$\displaystyle 3\,h^{3} = 1 - 9\,h^{3}$$

$$\displaystyle 3\,h^{3} + 9\,h^{3} = 1 - 9\,h^{3} + 9\,h^{3}$$

$$\displaystyle 12\,h^{3} = 1$$

$$\displaystyle 12(\dfrac{1}{12})\,h^{3} = 1(\dfrac{1}{12})$$

$$\displaystyle h^{3} = \dfrac{1}{12}$$

$$\displaystyle (h^{3})^{1/3} = (\dfrac{1}{12})^{1/3}$$

$$\displaystyle h = 0.4367902324$$ (one root)

Check

$$\displaystyle 15 + 3\,(0.4367902324)^{3} = 16 - 9\,(0.4367902324)^{3}$$

$$\displaystyle 15.25 \approx 15.25$$ (one root)

Yes

Find I. There should be three answers (roots).

$$\displaystyle 19 + 50\,i^{3} = 2 - \dfrac{4}{5}\,i^{3}$$

Last edited:

MarkFL

La Villa Strangiato
Staff member
Moderator
Math Helper
There are likely a complex conjugate pair of roots as well. A cubic equation (degree 3) will have 3 roots. Also, I wouldn't use the equal sign when subbing in a decimal approximations, but rather, I would use the LaTeX code:

\approx

TheJason

Staff member
Moderator
There are likely a complex conjugate pair of roots as well. A cubic equation (degree 3) will have 3 roots. Also, I wouldn't use the equal sign when subbing in a decimal approximations, but rather, I would use the LaTeX code:

\approx
Post edited. OK, we got one root at least. What would be the other two?

MarkFL

La Villa Strangiato
Staff member
Moderator
Math Helper
You could write the equation as:

$$\displaystyle 12h^3-1=0$$

Or

$$\displaystyle \left(\sqrt[3]{12}h\right)^3-1^3=0$$

Now, factor as the difference of cubes, and equate the two factors to zero to find all 3 roots.

TheJason

TheJason

Staff member
Moderator
You could write the equation as:

$$\displaystyle 12h^3-1=0$$

Or

$$\displaystyle \left(\sqrt[3]{12}h\right)^3-1^3=0$$

Now, factor as the difference of cubes, and equate the two factors to zero to find all 3 roots.
So in all cases of the cubed stuff, do you always use the $$\displaystyle -1 = 0\,$$ part? Why are those numbers used in this example?

Staff member