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9. 【阅读】有些大数值问题可以通过用字母代替数转化成整式问题来解决,请先阅读下面的解题过程,再解答后面的问题.
【先学】若$x = 123456789 × 123456786$,$y = 123456788 × 123456787$,试比较$x$,$y$的大小.
解:设$123456788 = a$,那么$x = (a + 1)(a - 2) = a^2 - a - 2$,$y = a(a - 1) = a^2 - a$,$\therefore x - y = (a^2 - a - 2) - (a^2 - a) = -2 < 0$,$\therefore x < y$.
【后练】
(1)已知$A = 999888321 × 123888999$,$B = 999888322 × 123888998$,试比较$A$,$B$的大小;
(2)计算:$3.456 × 2.456 × 5.456 - 3.456^3 - 1.456^2 - 4.456$.
答案:9.(1) 设 $999888321 = a$,$123888998 = b$,则 $A = a(b + 1) = ab + a$,$B = (a + 1)b = ab + b$,$\therefore A - B = ab + a - (ab + b) = a - b > 0$,$\therefore A > B$ (2) 设 $3.456 = a$,则原式 $= a(a - 1)(a + 2) - a^3 - (a - 2)^2 - (a + 1) = a^3 + a^2 - 2a - a^3 - a^2 + 4a - 4 - a - 1 = a - 5 = 3.456 - 5 = -1.544$
解析:
(1)设$999888321 = a$,$123888998 = b$,则$A = a(b + 1) = ab + a$,$B = (a + 1)b = ab + b$,$\therefore A - B = ab + a - (ab + b) = a - b$。因为$a = 999888321$,$b = 123888998$,所以$a - b>0$,$\therefore A>B$。
(2)设$3.456 = a$,则原式$= a(a - 1)(a + 2) - a^3 - (a - 2)^2 - (a + 1)$
$= a(a^2 + 2a - a - 2) - a^3 - (a^2 - 4a + 4) - a - 1$
$= a(a^2 + a - 2) - a^3 - a^2 + 4a - 4 - a - 1$
$= a^3 + a^2 - 2a - a^3 - a^2 + 4a - 4 - a - 1$
$= (a^3 - a^3) + (a^2 - a^2) + (-2a + 4a - a) + (-4 - 1)$
$= a - 5$
$= 3.456 - 5 = -1.544$
10. “$a^2 \geq 0$”这个结论在数学中非常有用,有时我们需要将代数式配成完全平方式.例如:$x^2 + 4x + 5 = x^2 + 4x + 4 + 1 = (x + 2)^2 + 1$,$\because (x + 2)^2 \geq 0$,$\therefore (x + 2)^2 + 1 \geq 1$,$\therefore x^2 + 4x + 5 \geq 1$.试利用“配方法”解决下面的问题:
(1)已知$x^2 - 4x + y^2 + 2y + 5 = 0$,求$x + y$的值;
(2)比较代数式$x^2 - 1$与$2x - 3$的大小.
答案:10.(1) $x^2 - 4x + y^2 + 2y + 5 = 0$ 可化为 $(x - 2)^2 + (y + 1)^2 = 0$。根据非负数的意义,得 $x - 2 = 0$,$y + 1 = 0$,解得 $x = 2$,$y = -1$。$\therefore x + y = 2 - 1 = 1$ (2) $x^2 - 1 - (2x - 3) = x^2 - 2x + 2 = (x - 1)^2 + 1$。$\because (x - 1)^2 \geq 0$,$\therefore (x - 1)^2 + 1 > 0$,$\therefore x^2 - 1 - (2x - 3) > 0$,$\therefore x^2 - 1 > 2x - 3$
解析:
(1)$x^2 - 4x + y^2 + 2y + 5 = 0$,可化为$(x - 2)^2 + (y + 1)^2 = 0$。因为$(x - 2)^2 \geq 0$,$(y + 1)^2 \geq 0$,所以$x - 2 = 0$,$y + 1 = 0$,解得$x = 2$,$y = -1$。则$x + y = 2 + (-1) = 1$。
(2)$x^2 - 1 - (2x - 3) = x^2 - 2x + 2 = (x - 1)^2 + 1$。因为$(x - 1)^2 \geq 0$,所以$(x - 1)^2 + 1 \geq 1 > 0$,即$x^2 - 1 - (2x - 3) > 0$,所以$x^2 - 1 > 2x - 3$。
11. (新考法·探究题)李老师在黑板上写了三个算式:①$3^2 - 1^2 = 8 × 1$;②$5^2 - 3^2 = 8 × 2$;③$7^2 - 5^2 = 8 × 3$.
(1)请结合上述三个算式的规律,写出第④个算式:
$9^2 - 7^2 = 8 × 4$

(2)设两个连续奇数为$2n - 1$,$2n + 1$(其中$n$为正整数),写出它们的平方差,并说明结果是8的倍数.
答案:11.(1) $9^2 - 7^2 = 8 × 4$ (2) $(2n + 1)^2 - (2n - 1)^2 = 4n^2 + 4n + 1 - (4n^2 - 4n + 1) = 4n^2 + 4n + 1 - 4n^2 + 4n - 1 = 8n$。$\because n$ 为正整数,$\therefore$ 结果是 $8$ 的倍数
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