![\frac{tg4x - tg3x}{1 + tg4x \times tg3x} = \sqrt{3} \\ \\ tg(4x - 3x) = \sqrt{3} \\ tgx = \sqrt{3} \\ x = \frac{\pi}{3} + \pi \: n](https://tex.z-dn.net/?f=+%5Cfrac%7Btg4x+-+tg3x%7D%7B1+%2B+tg4x+%5Ctimes+tg3x%7D++%3D++%5Csqrt%7B3%7D++%5C%5C++%5C%5C+tg%284x+-+3x%29+%3D++%5Csqrt%7B3%7D++%5C%5C+tgx+%3D++%5Csqrt%7B3%7D++%5C%5C+x+%3D++%5Cfrac%7B%5Cpi%7D%7B3%7D++%2B+%5Cpi+%5C%3A+n)
n € Z
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Воспользовались формулой:
![tg(x - y) = \frac{tgx - tgy}{1 + tgx \times tgy} \\](https://tex.z-dn.net/?f=tg%28x+-+y%29+%3D++%5Cfrac%7Btgx+-+tgy%7D%7B1+%2B+tgx+%5Ctimes+tgy%7D++%5C%5C+)
— тангенс разности
ОТВЕТ: х = π/3 + πn, n € Z
Для того, чтобы разложить квадратный трёхчлен, надо сначала решить соответствующее квадратное уравнение.
![9y^2+2y-7=0](https://tex.z-dn.net/?f=9y%5E2%2B2y-7%3D0+)
D=4+4*7*9=4+252=256
y₁=
![\frac{-2+ \sqrt{256} }{18} = \frac{-2+16}{18} = \frac{7}{9}](https://tex.z-dn.net/?f=+%5Cfrac%7B-2%2B+%5Csqrt%7B256%7D+%7D%7B18%7D+%3D+%5Cfrac%7B-2%2B16%7D%7B18%7D+%3D+%5Cfrac%7B7%7D%7B9%7D+)
y₂=
![\frac{-2-16}{18} = \frac{-18}{18} =-1](https://tex.z-dn.net/?f=+%5Cfrac%7B-2-16%7D%7B18%7D+%3D+%5Cfrac%7B-18%7D%7B18%7D+%3D-1)
Разложим квадратный трёхчлен:
![9y^2+2y-7=9(y+1)(y- \frac{7}{9} )=(y+1)(9y-7)](https://tex.z-dn.net/?f=9y%5E2%2B2y-7%3D9%28y%2B1%29%28y-+%5Cfrac%7B7%7D%7B9%7D+%29%3D%28y%2B1%29%289y-7%29)
![\dfrac{ \sqrt[3]{a^2\cdot \sqrt[4]{a^6\cdot a^{-1}} } }{\sqrt[4]{a^3\cdot \sqrt[3]{a^2} }} =\dfrac{ \sqrt[3]{a^2\cdot \sqrt[4]{a^{6-1}} } }{\sqrt[4]{a^3\cdot a^{\frac{2}{3}} }} =\dfrac{ \sqrt[3]{a^2\cdot a^{\frac{5}{4}} } }{\sqrt[4]{a^3\cdot a^{\frac{2}{3}} }} =\dfrac{ \sqrt[3]{a^{2+ \frac{5}{4} }} }{ \sqrt[4]{a^{3+ \frac{2}{3} }} }= \dfrac{ \sqrt[3]{a^{\frac{13}{4}}} }{ \sqrt[4]{a^{\frac{11}{3}}} } =\\\\\\=\dfrac{a^{\frac{13}{12}}}{a^{\frac{11}{12}}}=a^{\frac{13}{12}-\frac{11}{12}}=a^{\frac{1}{6}}](https://tex.z-dn.net/?f=+%5Cdfrac%7B+%5Csqrt%5B3%5D%7Ba%5E2%5Ccdot+%5Csqrt%5B4%5D%7Ba%5E6%5Ccdot+a%5E%7B-1%7D%7D+%7D+%7D%7B%5Csqrt%5B4%5D%7Ba%5E3%5Ccdot+%5Csqrt%5B3%5D%7Ba%5E2%7D+%7D%7D+%3D%5Cdfrac%7B+%5Csqrt%5B3%5D%7Ba%5E2%5Ccdot+%5Csqrt%5B4%5D%7Ba%5E%7B6-1%7D%7D+%7D+%7D%7B%5Csqrt%5B4%5D%7Ba%5E3%5Ccdot+a%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D+%7D%7D+%3D%5Cdfrac%7B+%5Csqrt%5B3%5D%7Ba%5E2%5Ccdot+a%5E%7B%5Cfrac%7B5%7D%7B4%7D%7D+%7D+%7D%7B%5Csqrt%5B4%5D%7Ba%5E3%5Ccdot+a%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D+%7D%7D+%3D%5Cdfrac%7B+%5Csqrt%5B3%5D%7Ba%5E%7B2%2B+%5Cfrac%7B5%7D%7B4%7D+%7D%7D+%7D%7B+%5Csqrt%5B4%5D%7Ba%5E%7B3%2B+%5Cfrac%7B2%7D%7B3%7D+%7D%7D+%7D%3D+%5Cdfrac%7B+%5Csqrt%5B3%5D%7Ba%5E%7B%5Cfrac%7B13%7D%7B4%7D%7D%7D+%7D%7B+%5Csqrt%5B4%5D%7Ba%5E%7B%5Cfrac%7B11%7D%7B3%7D%7D%7D+%7D+%3D%5C%5C%5C%5C%5C%5C%3D%5Cdfrac%7Ba%5E%7B%5Cfrac%7B13%7D%7B12%7D%7D%7D%7Ba%5E%7B%5Cfrac%7B11%7D%7B12%7D%7D%7D%3Da%5E%7B%5Cfrac%7B13%7D%7B12%7D-%5Cfrac%7B11%7D%7B12%7D%7D%3Da%5E%7B%5Cfrac%7B1%7D%7B6%7D%7D)
если все верно сделала, то ответ 1)
1) 4х+15-10х=5-11х
4х-10х+11х=5-15
5х=-10
х=-10:5
х=-2
2) 28у-42-5у-20=6у+10+5у
28у-5у-6у-5у=10+42+20
12у=72
у=72/12
у=6