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26. (10分)(2024·徐州鼓楼区期中)我国著名数学家华罗庚曾用诗词表达了“数形结合”的思想,其中谈到“数缺形时少直观,形少数时难入微.数形结合百般好,隔离分家万事休.”请你利用“数形结合”的思想解决以下问题:如图(1)是一个长为$4a$,宽为$b$的长方形,沿图中虚线用剪刀平均分成四块小长方形,然后用四块小长方形拼成如图(2)的图形.
(1)观察图形,写出一个$(a + b)^{2},(a - b)^{2},ab$三者之间的等量关系式: $(a + b)^{2}=(a - b)^{2}+4ab$;
(2)运用(1)中的结论,当$x - y = 7,xy = -6$时,求$x + y$的值;
(3)若$(m - 2023)(2024 - m)=-4$,求$(m - 2023)^{2}+(2024 - m)^{2}$的值.
解:(2)$\because x - y = 7,xy = -6,\therefore (x + y)^{2}=(x - y)^{2}+4xy = 7^{2}+4\times(-6)=49 - 24 = 25.\therefore x + y=\pm5$.
(3)设$m - 2023 = a,2024 - m = b$,则$a + b = 1,ab = -4$,
$\therefore (m - 2023)^{2}+(2024 - m)^{2}=a^{2}+b^{2}=(a + b)^{2}-2ab = 1^{2}-2\times(-4)=1 + 8 = 9$.
(1)观察图形,写出一个$(a + b)^{2},(a - b)^{2},ab$三者之间的等量关系式: $(a + b)^{2}=(a - b)^{2}+4ab$;
(2)运用(1)中的结论,当$x - y = 7,xy = -6$时,求$x + y$的值;
(3)若$(m - 2023)(2024 - m)=-4$,求$(m - 2023)^{2}+(2024 - m)^{2}$的值.
解:(2)$\because x - y = 7,xy = -6,\therefore (x + y)^{2}=(x - y)^{2}+4xy = 7^{2}+4\times(-6)=49 - 24 = 25.\therefore x + y=\pm5$.
(3)设$m - 2023 = a,2024 - m = b$,则$a + b = 1,ab = -4$,
$\therefore (m - 2023)^{2}+(2024 - m)^{2}=a^{2}+b^{2}=(a + b)^{2}-2ab = 1^{2}-2\times(-4)=1 + 8 = 9$.
答案：
26.
(1)(a + b)²=(a - b)² + 4ab
(2)
∵x - y = 7,xy = -6,
∴(x + y)²=(x - y)² + 4xy = 7² + 4×(-6)=49 - 24 = 25.
∴x + y = ±5.
(3)设m - 2023 = a,2024 - m = b,则a + b = 1,ab = -4,
∴(m - 2023)²+(2024 - m)²=a² + b²=(a + b)² - 2ab = 1² - 2×(-4)=1 + 8 = 9.
(1)(a + b)²=(a - b)² + 4ab
(2)
∵x - y = 7,xy = -6,
∴(x + y)²=(x - y)² + 4xy = 7² + 4×(-6)=49 - 24 = 25.
∴x + y = ±5.
(3)设m - 2023 = a,2024 - m = b,则a + b = 1,ab = -4,
∴(m - 2023)²+(2024 - m)²=a² + b²=(a + b)² - 2ab = 1² - 2×(-4)=1 + 8 = 9.
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