In mathematical expressions/equations, power to which a variable has been raised is called an exponent or index.

Any indicial equation which can be converted to a quadratic equation using proper substitution can be said to be in “quadratic-form”.

Equation given can be rearranged as:

\(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\)

\(\displaystyle\Leftrightarrow{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)

If we make the substitution:

\(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\)

The given equation can be transformed into quadratic equation in variable u as shown:

\(\displaystyle{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)

\(\displaystyle\Leftrightarrow{u}^{{2}}+{10}{u}+{9}={0}\)

Answer:

The equation \(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\) is said to be in quadratic form, because making the substitution \(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\) results in a new equation that is quadratic.

Any indicial equation which can be converted to a quadratic equation using proper substitution can be said to be in “quadratic-form”.

Equation given can be rearranged as:

\(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\)

\(\displaystyle\Leftrightarrow{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)

If we make the substitution:

\(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\)

The given equation can be transformed into quadratic equation in variable u as shown:

\(\displaystyle{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)

\(\displaystyle\Leftrightarrow{u}^{{2}}+{10}{u}+{9}={0}\)

Answer:

The equation \(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\) is said to be in quadratic form, because making the substitution \(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\) results in a new equation that is quadratic.