\(5a=10b=15c\)

hay \(\frac{a}{\frac{1}{5}}=\frac{b}{\frac{1}{10}}=\frac{c}{\frac{1}{15}}\)

\(\Rightarrow\frac{2a}{\frac{2}{5}}=\frac{3b}{\frac{3}{10}}=\frac{4c}{\frac{4}{15}}\)

Áp dụng tính chất của dãy tỉ số bằng nhau, ta có:

\(\frac{2a}{\frac{2}{5}}=\frac{3b}{\frac{3}{10}}=\frac{4c}{\frac{4}{15}}=\frac{2a+3b+4c}{\frac{2}{5}+\frac{3}{10}+\frac{4}{15}}=\frac{58}{\frac{29}{30}}=60\)

\(5a=60\Rightarrow a=\frac{60}{5}=12\)

\(10b=60\Rightarrow b=\frac{60}{10}=6\)

\(15c=60\Rightarrow c=\frac{60}{15}=4\)

con điên

\(5a^2+10b^2-6ab-4a+2b+3\)

\(=\left(a^2-6ab+9b^2\right)+\left(4a^2-4a+1\right)+\left(b^2+2b+1\right)+1\)

\(=\left(a-3b\right)^2+\left(2a-1\right)^2+\left(b+1\right)^2+1>0\left(đpcm\right)\)

câu b đề sai ak

\(A=\dfrac{ab+10b+25}{ab+5a+5b+25}+\dfrac{bc+10c+25}{bc+5b+5c+25}+\dfrac{ca+10a+25}{ac+5a+5c+25}\)

\(=\dfrac{\left(ab+5b\right)+\left(5b+25\right)}{\left(ab+5a\right)+\left(5b+25\right)}+\dfrac{\left(bc+5c\right)+\left(5c+25\right)}{\left(bc+5b\right)+\left(5c+25\right)}+\dfrac{\left(ca+5a\right)+\left(5a+25\right)}{\left(ac+5a\right)+\left(5c+25\right)}\)

\(=\dfrac{b\left(a+5\right)+5\left(b+5\right)}{a\left(b+5\right)+5\left(b+5\right)}+\dfrac{c\left(b+5\right)+5\left(c+5\right)}{b\left(c+5\right)+5\left(c+5\right)}+\dfrac{a\left(c+5\right)+5\left(a+5\right)}{a\left(c+5\right)+5\left(c+5\right)}\)

\(=\dfrac{b\left(a+5\right)+5\left(b+5\right)}{\left(a+5\right)\left(b+5\right)}+\dfrac{c\left(b+5\right)+5\left(c+5\right)}{\left(b+5\right)\left(c+5\right)}+\dfrac{a\left(c+5\right)+5\left(a+5\right)}{\left(a+5\right)\left(c+5\right)}\)

\(=\dfrac{b}{b+5}+\dfrac{5}{a+5}+\dfrac{c}{c+5}+\dfrac{5}{b+5}+\dfrac{a}{a+5}+\dfrac{5}{c+5}\)

\(=\left(\dfrac{b}{b+5}+\dfrac{5}{b+5}\right)+\left(\dfrac{a}{a+5}+\dfrac{5}{a+5}\right)+\left(\dfrac{c}{c+5}+\dfrac{5}{c+5}\right)\)

\(=1+1+1=3\) (\(a;b;c\ne-5\))

\(A=\dfrac{ab+5b+5b+25}{a\left(b+5\right)+5\left(b+5\right)}+\dfrac{bc+5c+5c+25}{b\left(c+5\right)+5\left(c+5\right)}+\dfrac{ca+5a+5a+25}{a\left(c+5\right)+5\left(c+5\right)}\)

\(A=\dfrac{b\left(a+5\right)+5\left(b+5\right)}{\left(a+5\right)\left(b+5\right)}+\dfrac{c\left(b+5\right)+5\left(c+5\right)}{\left(b+5\right)\left(c+5\right)}+\dfrac{a\left(c+5\right)+5\left(a+5\right)}{\left(a+5\right)\left(c+5\right)}\)

\(A=\dfrac{b}{b+5}+\dfrac{5}{a+5}+\dfrac{c}{c+5}+\dfrac{5}{b+5}+\dfrac{a}{a+5}+\dfrac{5}{c+5}\)

\(A=\dfrac{a+5}{a+5}+\dfrac{b+5}{b+5}+\dfrac{c+5}{c+5}=1+1+1=3\)