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

Your Turn

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

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

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

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

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

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

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

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

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

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

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

Check

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

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

Yes

Your Turn

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

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

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