0\\ x+24>0 \end{cases}\Rightarrow \begin{cases} x>-3\\ x>-24 \end{cases}\Rightarrow x>-3\\ \frac12\lg(x+3)=\frac12\lg100-\frac12\lg(x+24)\\ \lg(x+3)=\lg100-\lg(x+24)\\ \lg(x+3)=\lg\frac{100}{x+24}\\ x+3=\frac{100}{x+24}\\ (x+3)(x+24)=100\\ x^2+27x+72=100\\ x^2+27x-28=0\\ D=729+4\cdot28=841=(29)^2\\ x_1=1,\;x_2=-28\;-\;HE\;nogx.\;no\;O.O.\Phi\\ x=1" alt="1.\;\frac12\lg(x+3)=1-\frac12\lg(x+24)\\ O.O.\Phi:\\ \begin{cases} x+3>0\\ x+24>0 \end{cases}\Rightarrow \begin{cases} x>-3\\ x>-24 \end{cases}\Rightarrow x>-3\\ \frac12\lg(x+3)=\frac12\lg100-\frac12\lg(x+24)\\ \lg(x+3)=\lg100-\lg(x+24)\\ \lg(x+3)=\lg\frac{100}{x+24}\\ x+3=\frac{100}{x+24}\\ (x+3)(x+24)=100\\ x^2+27x+72=100\\ x^2+27x-28=0\\ D=729+4\cdot28=841=(29)^2\\ x_1=1,\;x_2=-28\;-\;HE\;nogx.\;no\;O.O.\Phi\\ x=1" align="absmiddle" class="latex-formula">
2![2.\;2\sin^23x+\sin^26x=2\\ 2\sin^23x+4\sin^23x\cos^23x=2\sin^23x+2\cos^23x\\ 4\sin^23x\cos^23x-2\cos^23x=0\\ 2\cos^23x(2\sin^23x-1)=0\\ 2\cos^23x=0\;-\;HA\;[0;\frac\pi4]\;pew.\;HET\\ 2\sin^23x-1=0\\ \sin^23x=\frac12\\ \sin3x=\frac1{\sqrt2}\\ 3x=\frac\pi4+2\pi n\\ x=\frac\pi{12}+\frac{2\pi}3n,\;n\in\mathbb{Z}\\ 0\leq\frac\pi{12}+\frac{2\pi}3n\leq\frac\pi4\\ -\frac\pi{12}\leq\frac{2\pi}3n\leq\frac\pi6\\ -\frac18\leq n\leq\frac14\\ n\in\mathbb{Z}\Rightarrow n=0\\ x=\frac\pi{12}](https://tex.z-dn.net/?f=2.%5C%3B2%5Csin%5E23x%2B%5Csin%5E26x%3D2%5C%5C+2%5Csin%5E23x%2B4%5Csin%5E23x%5Ccos%5E23x%3D2%5Csin%5E23x%2B2%5Ccos%5E23x%5C%5C+4%5Csin%5E23x%5Ccos%5E23x-2%5Ccos%5E23x%3D0%5C%5C+2%5Ccos%5E23x%282%5Csin%5E23x-1%29%3D0%5C%5C+2%5Ccos%5E23x%3D0%5C%3B-%5C%3BHA%5C%3B%5B0%3B%5Cfrac%5Cpi4%5D%5C%3Bpew.%5C%3BHET%5C%5C+2%5Csin%5E23x-1%3D0%5C%5C+%5Csin%5E23x%3D%5Cfrac12%5C%5C+%5Csin3x%3D%5Cfrac1%7B%5Csqrt2%7D%5C%5C+3x%3D%5Cfrac%5Cpi4%2B2%5Cpi+n%5C%5C+x%3D%5Cfrac%5Cpi%7B12%7D%2B%5Cfrac%7B2%5Cpi%7D3n%2C%5C%3Bn%5Cin%5Cmathbb%7BZ%7D%5C%5C+0%5Cleq%5Cfrac%5Cpi%7B12%7D%2B%5Cfrac%7B2%5Cpi%7D3n%5Cleq%5Cfrac%5Cpi4%5C%5C+-%5Cfrac%5Cpi%7B12%7D%5Cleq%5Cfrac%7B2%5Cpi%7D3n%5Cleq%5Cfrac%5Cpi6%5C%5C+-%5Cfrac18%5Cleq+n%5Cleq%5Cfrac14%5C%5C+n%5Cin%5Cmathbb%7BZ%7D%5CRightarrow+n%3D0%5C%5C+x%3D%5Cfrac%5Cpi%7B12%7D "2.\;2\sin^23x+\sin^26x=2\\ 2\sin^23x+4\sin^23x\cos^23x=2\sin^23x+2\cos^23x\\ 4\sin^23x\cos^23x-2\cos^23x=0\\ 2\cos^23x(2\sin^23x-1)=0\\ 2\cos^23x=0\;-\;HA\;[0;\frac\pi4]\;pew.\;HET\\ 2\sin^23x-1=0\\ \sin^23x=\frac12\\ \sin3x=\frac1{\sqrt2}\\ 3x=\frac\pi4+2\pi n\\ x=\frac\pi{12}+\frac{2\pi}3n,\;n\in\mathbb{Z}\\ 0\leq\frac\pi{12}+\frac{2\pi}3n\leq\frac\pi4\\ -\frac\pi{12}\leq\frac{2\pi}3n\leq\frac\pi6\\ -\frac18\leq n\leq\frac14\\ n\in\mathbb{Z}\Rightarrow n=0\\ x=\frac\pi{12}")
Уравнение имеет один корень на заданном отрезке.
3. Числитель x, знаменатель (x+3).
