Kết quả:

1. \(-\frac{2}{3}\)

2. \(3\)

1.

        \(\cos2x+\sin\left(x+\frac{pi}{4}\right)=0\)

\(\Leftrightarrow\sin\left(x+\frac{pi}{4}\right)=-\cos2x\)

\(\Leftrightarrow\sin\left(x+\frac{pi}{4}\right)=\sin\left(2x-\frac{pi}{2}\right)\)

\(\Leftrightarrow\orbr{\begin{cases}x+\frac{pi}{4}=2x-\frac{pi}{2}+k2pi\\x+\frac{pi}{4}=pi-2x+\frac{pi}{2}+k2pi\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}-x=-\frac{3}{4}pi+k2pi\\3x=+\frac{5}{4}pi+k2pi\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}x=\frac{3}{4}pi+k2pi\\x=\frac{5}{12}pi+k\frac{2}{3}pi\end{cases}}\)

2.

\(\sin\left(3x-\frac{5pi}{6}\right)+\cos\left(3x+\frac{3pi}{6}\right)=0\)

\(\Leftrightarrow\sin\left(3x-\frac{5pi}{6}\right)=-\cos\left(3x+\frac{3pi}{6}\right)\)

\(\Leftrightarrow\sin\left(3x-\frac{5pi}{6}\right)=\sin\left(3x+\frac{3pi}{6}-\frac{pi}{2}\right)\)

\(\Leftrightarrow\orbr{\begin{cases}3x-\frac{5pi}{6}=3x+\frac{3pi}{6}-\frac{pi}{2}+k2pi\\3x-\frac{5pi}{6}=pi-3x-\frac{3pi}{6}+\frac{pi}{2}+k2pi\end{cases}}\)

\(\Leftrightarrow\orbr{\begin{cases}0x=\frac{5pi}{6}+k2pi\left(VN\right)\\6x=\frac{11pi}{6}+k2pi\end{cases}}\)

\(\Leftrightarrow x=\frac{11pi}{36}+k\frac{1}{3}pi\)

Thiếu vế phải rồi bạn

Sorry bn tai vua nay no bi loi

\(\Leftrightarrow-\frac{4x}{x^2-5x+6}-\frac{3x}{x^2-7x+6}+6=0\)

\(\Rightarrow\frac{6x^4-79x^3+325x^2-474x+216}{\left(x^2-7x+6\right)\left(x^2-5x+6\right)}=0\)

=>6x⁴-79x³+325x²-474x+216=0

denta:3x²-23x+18=0

=>(-23)²-4(3.18)=313

\(\Rightarrow x_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{23\pm\sqrt{313}}{6}\)

=>x=4;\(\frac{3}{2};\frac{\sqrt{313}}{6}+3\frac{5}{6};3\frac{5}{6}-\frac{\sqrt{313}}{6}\)

vậy pt trên có 4 nghiệm

\(\frac{4x}{x^2-5x+6}+\frac{3x}{x^2-7x+6}=6\)

\(\Leftrightarrow\frac{4}{x-5+\frac{6}{x}}+\frac{3}{x-7+\frac{6}{x}}=6\)

Đặt \(x+\frac{6}{x}=a\)

\(\Leftrightarrow\frac{4}{a-5}+\frac{3}{a-7}=6\)

\(\Leftrightarrow\frac{4a-28}{\left(a-5\right)\left(a-7\right)}+\frac{3a-15}{\left(a-5\right)\left(a-7\right)}=6\)

\(\Leftrightarrow4a-28+3a-15=6\left(a^2-12a+35\right)\)

\(\Leftrightarrow6a^2-79a+253=0\)

\(\Delta=79^2-4.253.6=169>0\)

=> Phương trình có 2 nghiệm phân biệt 

\(x_1=\frac{79+13}{2.6}=\frac{23}{3}\)( nhận)

\(x_2=\frac{79-13}{12}=5.5\) ( nhận)

Vậy số nghiệm của phương trình là 2 nghiệm

k mình nha!!!!

có 4 nghiệm