Question

Tìm các số hữu tỉ a,b,c biết:

a, ab = \(\dfrac{3}{5},bc=\dfrac{4}{5},ca=\dfrac{3}{4}\)

b, a.(a + b + c) = -12 và b.(a + b + c) = 18 và c.(a + b + c) = 30

c, ab = c, bc = 4a, ac = 9b

Answer 1

a) \(ab=\dfrac{3}{5};bc=\dfrac{4}{5};ca=\dfrac{3}{4}\)

\(\Leftrightarrow ab.bc.ca=\dfrac{3}{5}.\dfrac{4}{5}.\dfrac{3}{4}\)

\(\Leftrightarrow a^2.b^2.c^2=\dfrac{9}{25}\)

\(\Leftrightarrow\left(abc\right)^2=\left(\dfrac{3}{5}\right)^2=\left(-\dfrac{3}{5}\right)^2\)

+ Khi \(\left(abc\right)^2=\left(\dfrac{3}{5}\right)^2\Leftrightarrow abc=\dfrac{3}{5}\)

Vậy \(\left\{{}\begin{matrix}a=\dfrac{3}{5}:\dfrac{4}{5}=\dfrac{3}{4}\\b=\dfrac{3}{5}:\dfrac{3}{4}=\dfrac{4}{5}\\c=\dfrac{3}{5}:\dfrac{3}{5}=1\end{matrix}\right.\)

+ Khi \(\left(abc\right)^2=\left(-\dfrac{3}{5}\right)^2\Leftrightarrow abc=-\dfrac{3}{5}\)

Vậy \(\left\{{}\begin{matrix}a=\left(-\dfrac{3}{5}\right):\dfrac{4}{5}=-\dfrac{3}{4}\\b=\left(-\dfrac{3}{5}\right):\dfrac{3}{4}=-\dfrac{4}{5}\\c=\left(-\dfrac{3}{5}\right):\dfrac{3}{5}=-1\end{matrix}\right.\)

b) \(a\left(a+b+c\right)=-12;b\left(a+b+c\right)=18;c\left(a+b+c\right)=30\)

\(\Leftrightarrow a\left(a+b+c\right)+b\left(a+b+c\right)+c\left(a+b+c\right)=\left(-12\right)+18+30\)

\(\Leftrightarrow\left(a+b+c\right)\left(a+b+c\right)=36\)

\(\Leftrightarrow\left(a+b+c\right)^2=6^2=\left(-6\right)^2\)

+ Khi \(\left(a+b+c\right)^2=6^2\Leftrightarrow a+b+c=6\)

Vậy \(\left\{{}\begin{matrix}a=\left(-12\right):6=-2\\b=18:6=3\\c=30:6=5\end{matrix}\right.\)

+ Khi \(\left(a+b+c\right)^2=\left(-6\right)^2\Leftrightarrow a+b+c=-6\)

Vậy \(\left\{{}\begin{matrix}a=\left(-12\right):\left(-6\right)=2\\b=18:\left(-6\right)=-3\\c=30:\left(-6\right)=-5\end{matrix}\right.\)

c) \(ab=c;bc=4a;ac=9b\)

Kiểm tra lại đề bài xem có thiếu điều kiện không.

Answer 2

Cứ theo khẳng định của Nguyễn Thị Ngọc Linh thì đề c) không thiếu gì. Xin giải tiếp.

c) \(ab=c;bc=4a;ac=9b\)

\(\Leftrightarrow ab.bc.ac=c.4a.9b\)

\(\Leftrightarrow\left(abc\right)\left(abc\right)=36\left(abc\right)\)

\(\Leftrightarrow abc=36\)

+ Vì \(ab=c\Leftrightarrow cc=36\Leftrightarrow c^2=6^2=\left(-6\right)^2\)

+ Vì \(bc=4a\Leftrightarrow a.4a=36\Leftrightarrow4a^2=36\Leftrightarrow a^2=9=3^2=\left(-3\right)^2\)

+ Vì \(ac=9b\Leftrightarrow b.9b=36\Leftrightarrow9b^2=36\Leftrightarrow b^2=4=2^2=\left(-2\right)^2\)

Vậy \(\left\{{}\begin{matrix}a_1=3;a_2=-3\\b_1=2;b_2=-2\\c_1=6;c_2=-6\end{matrix}\right.\)