题目

Problem

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve $C_1$ has equation

y=x^4+10x^2+8\qquad x\in\mathbb{R}

The curve $C_2$ has equation

y=2x^2-7\qquad x\in\mathbb{R}

Use algebra to prove by contradiction that $C_1$ and $C_2$ do not intersect.

(4)

题目中文翻译

本题必须写出全部解题步骤。

不接受依赖计算器技术的解法。

曲线 $C_1$ 的方程为

y=x^4+10x^2+8\qquad x\in\mathbb{R}

曲线 $C_2$ 的方程为

y=2x^2-7\qquad x\in\mathbb{R}

用代数中的反证法证明 $C_1$ 与 $C_2$ 不相交。

解答

Assume, for contradiction, that $C_1$ and $C_2$ intersect.

Then there is a real value of $x$ for which the two $y$ -values are equal, so

x^4+10x^2+8=2x^2-7

Rearrange:

x^4+8x^2+15=0

Factorise:

x^4+8x^2+15=(x^2+3)(x^2+5)

(x^2+3)(x^2+5)=0

This gives

x^2=-3

x^2=-5

But for real $x$ ,

x^2\ge 0

So $x^2$ cannot be negative.

This is a contradiction.

Therefore the assumption that $C_1$ and $C_2$ intersect is false.

Hence

\boxed{C_1\text{ and }C_2\text{ do not intersect.}}