# The equation m^(2/3) + 10m^(1/3) + 9 = 0 is said to be in __________form, because making the substitution u = __________results in a new equation that is quadratic.

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

2020-10-24
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:
$$\displaystyle{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}$$
$$\displaystyle\Leftrightarrow{u}^{{2}}+{10}{u}+{9}={0}$$
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.

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