Question

solve with quadratic formula
${x}^{2 \div 5 } - 5x {}^{1 \div 5} + 6 = 0$
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Answer 1

Given to solve,

$\longrightarrow\sf{x^{\frac{2}{5}}-5x^{\frac{1}{5}}+6=0}$

$\longrightarrow\sf{x^{\frac{1}{5}\times2}-5x^{\frac{1}{5}}+6=0}$

$\longrightarrow\sf{\left(x^{\frac{1}{5}}\right)^2-5\left(x^{\frac{1}{5}}\right)+6=0}$

Let $\sf{k=x^{\frac{1}{5}}.}$ Then the equation becomes,

$\longrightarrow\sf{k^2-5k+6=0}$

Here,

• $\sf{a=1}$

• $\sf{b=-5}$

• $\sf{c=6}$

Solving for $\sf{k}$ by applying quadratic formula,

$\longrightarrow\sf{k=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}}$

$\longrightarrow\sf{k=\dfrac{5\pm\sqrt{(-5)^2-4\times1\times6}}{2\times1}}$

$\longrightarrow\sf{k=\dfrac{5\pm\sqrt{25-24}}{2}}$

$\longrightarrow\sf{k=\dfrac{5\pm\sqrt{1}}{2}}$

$\longrightarrow\sf{k=\dfrac{5\pm1}{2}}$

$\longrightarrow\sf{k=\dfrac{5+1}{2}\quad OR\quad k=\dfrac{5-1}{2}}$

$\longrightarrow\sf{k=3\quad OR\quad k=2}$

Undoing $\sf{k=x^{\frac{1}{5}},}$

$\longrightarrow\sf{x^{\frac{1}{5}}=3\quad OR\quad x^{\frac{1}{5}}=2}$

$\longrightarrow\sf{x=3^5\quad OR\quad x=2^5}$

$\longrightarrow\sf{\underline{\underline{x=243\quad OR\quad x=32}}}$

Answer 2

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