化简 $(a+2b)(a-2b)-a^2$。 答案 ... 由此计算 $12\,349\times12\,341-12\,345^2$ 的值。 答案 ...

1. Recognise and expand the product using difference of two squares

$(a+2b)(a-2b)=a^2-(2b)^2=a^2-4b^2$

2. Subtract $a^2$ and collect like terms

$(a+2b)(a-2b)-a^2=a^2-4b^2-a^2=-4b^2$

1. Identify the values of $a$ and $b$ by comparing with the expression from part (a)

From part (a),
$(a+2b)(a-2b)-a^2=-4b^2.$
Compare
$12\,349\times 12\,341-12\,345^2$
with
$(a+2b)(a-2b)-a^2.$
So
$a=12\,345,$
and
$a+2b=12\,349,\quad a-2b=12\,341.$
Hence
$2b=4 \quad \Rightarrow \quad b=2.$

2. Substitute $b=2$ into the simplified result from part (a)

Using $-4b^2$,
$-4(2)^2=-4\times 4=-16.$