Question

найдите все мары целых чисел n и m, для которых выполняется равенство n^2-m^2=2m-n-9.​

Answer 1

Ответ:

$(9;-10),\ (3;-3),\ (3;2),\ (9;9),\ (-11;-10),\ (-5;-3),\ (-5;2),\ (-11;9)$

Пошаговое объяснение:

$n^2-m^2=2m-n-9$

$n^2+n-m^2-2m+9=0$

$n^2+n+\frac{1}{4}-m^2-2m-1+9\frac{3}{4}=0$

$\left( n+\frac{1}{2}\right)^2-(m+1)^2+9\frac{3}{4}=0$

$\left( n+\frac{1}{2}\right)^2-(m+1)^2=-\frac{39}{4}\ \ \ |\cdot 4$

$\left(2n+1\right)^2-(2m+2)^2=-39$

$(2n+1-2m-2)(2n+1+2m+2)=-39$

$(2n-2m-1)(2n+2m+3)=-39$

1.

$(2n-2m-1)(2n+2m+3)=-39\cdot 1$

$\begin{cases}2n-2m-1=-39\\2n+2m+3=1 \end{cases}$

$\begin{cases}2n-2m=-39+1\\2n+2m=1-3 \end{cases}$

$\begin{cases}2n-2m=-38\\2n+2m=-2 \end{cases}$

+_________________

$4n=-40\ \ \ |:4$

$n=-10$

$2n+2m=-2$

$2\cdot(-10)+2m=-2$

$-20+2m=-2$

$2m=-2+20$

$2m=18\ \ \ |:2$

$m=9$

$(9;-10)$

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2.

$(2n-2m-1)(2n+2m+3)=-13\cdot 3$

$\begin{cases}2n-2m-1=-13\\2n+2m+3=3 \end{cases}$

$\begin{cases}2n-2m=-13+1\\2n+2m=3-3 \end{cases}$

$\begin{cases}2n-2m=-12\\2n+2m=0 \end{cases}$

+_________________

$4n=-12\ \ \ |:4$

$n=-3$

$2n+2m=0$

$2\cdot(-3)+2m=0$

$-6+2m=0$

$2m=0+6$

$2m=6\ \ \ |:2$

$m=3$

$(3;-3)$

==============================

3.

$(2n-2m-1)(2n+2m+3)=-3\cdot 13$

$\begin{cases}2n-2m-1=-3\\2n+2m+3=13 \end{cases}$

$\begin{cases}2n-2m=-3+1\\2n+2m=13-3 \end{cases}$

$\begin{cases}2n-2m=-2\\2n+2m=10 \end{cases}$

+_________________

$4n=8\ \ \ |:4$

$n=2$

$2n+2m=10$

$2\cdot2+2m=10$

$4+2m=10$

$2m=10-4$

$2m=6\ \ \ |:2$

$m=3$

$(3;2)$

==============================

4.

$(2n-2m-1)(2n+2m+3)=-1\cdot 39$

$\begin{cases}2n-2m-1=-1\\2n+2m+3=39 \end{cases}$

$\begin{cases}2n-2m=-1+1\\2n+2m=39-3 \end{cases}$

$\begin{cases}2n-2m=0\\2n+2m=36 \end{cases}$

+_________________

$4n=36\ \ \ |:4$

$n=9$

$2n+2m=36$

$2\cdot9+2m=36$

$18+2m=36$

$2m=36-18$

$2m=18\ \ \ |:2$

$m=9$

$(9;9)$

==============================

5.

$(2n-2m-1)(2n+2m+3)=1\cdot (-39)$

$\begin{cases}2n-2m-1=1\\2n+2m+3=-39 \end{cases}$

$\begin{cases}2n-2m=1+1\\2n+2m=-39-3 \end{cases}$

$\begin{cases}2n-2m=2\\2n+2m=-42 \end{cases}$

+_________________

$4n=-40\ \ \ |:4$

$n=-10$

$2n+2m=-42$

$2\cdot(-10)+2m=-42$

$-20+2m=-42$

$2m=-42+20$

$2m=-22\ \ \ |:2$

$m=-11$

$(-11;-10)$

==============================

6.

$(2n-2m-1)(2n+2m+3)=3\cdot (-13)$

$\begin{cases}2n-2m-1=3\\2n+2m+3=-13 \end{cases}$

$\begin{cases}2n-2m=3+1\\2n+2m=-13-3 \end{cases}$

$\begin{cases}2n-2m=4\\2n+2m=-16 \end{cases}$

+_________________

$4n=-12\ \ \ |:4$

$n=-3$

$2n+2m=-16$

$2\cdot(-3)+2m=-16$

$-6+2m=-16$

$2m=-16+6$

$2m=-10\ \ \ |:2$

$m=-5$

$(-5;-3)$

==============================

7.

$(2n-2m-1)(2n+2m+3)=13\cdot (-3)$

$\begin{cases}2n-2m-1=13\\2n+2m+3=-3 \end{cases}$

$\begin{cases}2n-2m=13+1\\2n+2m=-3-3 \end{cases}$

$\begin{cases}2n-2m=14\\2n+2m=-6 \end{cases}$

+_________________

$4n=8\ \ \ |:4$

$n=2$

$2n+2m=-6$

$2\cdot2+2m=-6$

$4+2m=-6$

$2m=-6-4$

$2m=-10\ \ \ |:2$

$m=-5$

$(-5;2)$

==============================

8.

$(2n-2m-1)(2n+2m+3)=39\cdot (-1)$

$\begin{cases}2n-2m-1=39\\2n+2m+3=-1 \end{cases}$

$\begin{cases}2n-2m=39+1\\2n+2m=-1-3 \end{cases}$

$\begin{cases}2n-2m=40\\2n+2m=-4 \end{cases}$

+_________________

$4n=36\ \ \ |:4$

$n=9$

$2n+2m=-4$

$2\cdot9+2m=-4$

$18+2m=-4$$2m=-4-18$

$2m=-22\ \ \ |:2$

$m=-11$

$(-11;9)$