Tìm số tự nhiên a,b,c biết:
1) a+b=17,a+c=15,a+b+c=20
2) a+b=5,b+c=9,a+c=6
3) ab=24,bc=96,abc=288
4) ab=36,bc=45,ca=20
5) a+b=7,b+c=9,c+a=8
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TQ
giải giúp mình hệ phương trình 3 ẩn này với mình cảm ơn nhiều
3=(a+b)(a+c)
4=(a+b)(b+c)
5=(a+c)(b+c)
1
NL
Nguyễn Lê Phước Thịnh
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10 tháng 10
TA có: \(\begin{cases} (a+b)(a+c) = 3 \quad (1) \\ (a+b)(b+c) = 4 \quad (2) \\ (a+c)(b+c) = 5 \quad (3) \end{cases}\)
=>\([(a+b)(b+c)(a+c)]^2 = 3 \times 4 \times 5 = 60\)
=>\((a+b)(b+c)(a+c) = \pm\sqrt{60} = \pm 2\sqrt{15}\)
Vì (a+b)(a+c)=3>0; (a+b)(b+c)=4>0; (a+c)(b+c)=5>0
nen các tổng a+b; a+c; b+c phải cùng dương hoặc cùng âm
TH1: a+b>0; a+c>0; b+c>0
Do đó, ta có: \(\left(a+b\right)\left(b+c\right)\left(a+c\right)=2\sqrt{15}\)
=>\(\begin{cases}b+c=\frac{2\sqrt{15}}{3}\\ a+c=\frac{2\sqrt{15}}{4}=\frac{\sqrt{15}}{2}\\ a+b=\frac{2\sqrt{15}}{5}\end{cases}\)
=>\((a+b) + (b+c) + (a+c) = \frac{2\sqrt{15}}{3} + \frac{\sqrt{15}}{2} + \frac{2\sqrt{15}}{5}\)
=>\(2(a+b+c) = \sqrt{15}\left(\frac{2}{3} + \frac{1}{2} + \frac{2}{5}\right) = \sqrt{15} \cdot \frac{47}{30} = \frac{47\sqrt{15}}{30}\)
\(\implies a+b+c = \frac{47\sqrt{15}}{60}\)
Do đó: \(\begin{cases}a=(a+b+c)-(b+c)=\frac{47\sqrt{15}}{60}-\frac{2\sqrt{15}}{3}=\frac{7\sqrt{15}}{60}\\ b=(a+b+c)-(a+c)=\frac{47\sqrt{15}}{60}-\frac{\sqrt{15}}{2}=\frac{17\sqrt{15}}{60}\\ c=(a+b+c)-(a+b)=\frac{47\sqrt{15}}{60}-\frac{2\sqrt{15}}{5}=\frac{23\sqrt{15}}{60}\end{cases}\)
TH2: a+b<0; a+c<0; b+c<0
Do đó, ta có: \(\left(a+b\right)\left(b+c\right)\left(a+c\right)=-2\sqrt{15}\)
=>\(\begin{cases}b+c=\frac{-2\sqrt{15}}{3}\\ a+c=\frac{2\sqrt{15}}{4}=\frac{-\sqrt{15}}{2}\\ a+b=\frac{-2\sqrt{15}}{5}\end{cases}\)
=>\((a+b)+(b+c)+(a+c)=\frac{-2\sqrt{15}}{3}+\frac{-\sqrt{15}}{2}+\frac{-2\sqrt{15}}{5}\)
=>\(2(a+b+c)=-\sqrt{15}\left(\frac{2}{3}+\frac{1}{2}+\frac{2}{5}\right)=\sqrt{15}\cdot\frac{-47}{30}=\frac{-47\sqrt{15}}{30}\)
=>\(a+b+c=-\frac{47\sqrt{15}}{60}\)
=>\(\begin{cases}a=(a+b+c)-(b+c)=-\frac{47\sqrt{15}}{60}-\left(-\frac{2\sqrt{15}}{3}\right)=-\frac{47\sqrt{15}}{60}+\frac{40\sqrt{15}}{60}=-\frac{7\sqrt{15}}{60}\\ b=(a+b+c)-(a+c)=-\frac{47\sqrt{15}}{60}-\left(-\frac{\sqrt{15}}{2}\right)=-\frac{47\sqrt{15}}{60}+\frac{30\sqrt{15}}{60}=-\frac{17\sqrt{15}}{60}\\ c=(a+b+c)-(a+b)=-\frac{47\sqrt{15}}{60}-\left(-\frac{2\sqrt{15}}{5}\right)=-\frac{47\sqrt{15}}{60}+\frac{24\sqrt{15}}{60}=-\frac{23\sqrt{15}}{60}\end{cases}\)
9 tháng 1 2023
ĐKXĐ: \(x\ge-1\)
Đặt \(\sqrt{x+1}=y\ge0\)
\(\Rightarrow4x^2+12xy=27y^2\)
\(\Leftrightarrow\left(2x-3y\right)\left(2x+9y\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}3y=2x\\9y=-2x\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3\sqrt{x+1}=2x\left(x\ge0\right)\\9\sqrt{x+1}=-2x\left(x\le0\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}9\left(x+1\right)=4x^2\left(x\ge0\right)\\81\left(x+1\right)=4x^2\left(x\le0\right)\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=3\\x=\dfrac{81-9\sqrt{97}}{8}\end{matrix}\right.\)
5: a+b=7
=>a=7-b
b+c=9
=>c=9-b
c+a=8
=>7-b+9-b=8
=>16-2b=8
=>2b=16-8=8
=>b=4
=>a=7-4=3;c=9-4=5
1: a+b=17
a+b+c=20
=>c=20-17=3
a+c=15
=>a=15-c=15-3=1
b=17-a=17-1=16
2: \(\left\{{}\begin{matrix}a+b=5\\b+c=9\\a+c=6\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=5-b\\c=9-b\\5-b+9-b=6\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}14-2b=6\\a=5-b\\c=9-b\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2b=8\\a=5-b\\c=9-b\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b=4\\a=5-4=1\\c=9-4=5\end{matrix}\right.\)
3: \(c=\dfrac{abc}{ab}=\dfrac{288}{24}=12\)
bc=96
=>b=96/12=8
\(a=\dfrac{24}{b}=\dfrac{24}{8}=3\)
4: \(\left\{{}\begin{matrix}ab=36\\bc=45\\ca=20\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=\dfrac{36}{b}\\c=\dfrac{45}{b}\\\dfrac{36}{b}\cdot\dfrac{45}{b}=20\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}b^2=36\cdot45:20=81\\a=\dfrac{36}{b}\\c=\dfrac{45}{b}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}b\in\left\{9;-9\right\}\\a=\dfrac{36}{b}\\c=\dfrac{45}{b}\end{matrix}\right.\)
TH1: b=9
\(a=\dfrac{36}{9}=4;c=\dfrac{45}{b}=\dfrac{45}{9}=5\)
TH2: b=-9
=>\(a=\dfrac{36}{-9}=-4;c=\dfrac{45}{-9}=-5\)