题目

Problem

Solve, using algebra, the equation

\begin{align*} x-6x^{1/2}+4=0. \end{align*}

Fully simplify your answers, writing them in the form $a+b\sqrt{c}$ , where $a$ , $b$ and $c$ are integers to be found.

(5)

解答

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令 $u=\sqrt{x}$ ，则 $x=u^2$ 。先解关于 $u$ 的二次方程，再把 $u$ 的值平方回到 $x$ 。

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Let

\begin{align*} u=x^{1/2}. \end{align*}

Then $x=u^2$ , so

\begin{align*} u^2-6u+4=&\,0. \end{align*}

Using the quadratic formula,

\begin{align*} u=&\,\frac{6\pm\sqrt{36-16}}{2}\\ =&\,\frac{6\pm\sqrt{20}}{2}\\ =&\,3\pm\sqrt5. \end{align*}

Therefore

\begin{align*} x=&\,(3\pm\sqrt5)^2\\ =&\,9\pm6\sqrt5+5\\ =&\,14\pm6\sqrt5. \end{align*}

\begin{align*} x=14+6\sqrt5 \quad\text{or}\quad x=14-6\sqrt5. \end{align*}