4х + 25/13 = 5 (умножим обе части на 13)
52х + 25 = 65
52х = 65 - 25
52х = 40
х = 40/52
х = 10/13
X-2,5*x>2-5
-1,5*x > -3
1,5*x < 3
x <2
1) S=4πR²=36π (см²);
R²=36π/(4π)=9;
R=3 см.
Ответ: 3 см.
2)
![log_{ \sqrt{5}}25= log_{5^{ \frac{1}{2}}}25=2 log_{5}25=2*2=4. \\](https://tex.z-dn.net/?f=+log_%7B+%5Csqrt%7B5%7D%7D25%3D+log_%7B5%5E%7B+%5Cfrac%7B1%7D%7B2%7D%7D%7D25%3D2+log_%7B5%7D25%3D2%2A2%3D4.+%5C%5C++++)
Ответ: 4.
![3) 7^{3-x}\ \textgreater \ \frac{1}{49}; \\ 7^{3-x}\ \textgreater \ 7^{-2}; \\ 3-x\ \textgreater \ -2; \\ -x\ \textgreater \ -2-3; \\ -x\ \textgreater \ -5; \\ x\ \textless \ 5. \\ ](https://tex.z-dn.net/?f=3%29+7%5E%7B3-x%7D%5C+%5Ctextgreater+%5C+%5Cfrac%7B1%7D%7B49%7D%3B+%5C%5C+%0A7%5E%7B3-x%7D%5C+%5Ctextgreater+%5C+7%5E%7B-2%7D%3B+%5C%5C+%0A3-x%5C+%5Ctextgreater+%5C+-2%3B+%5C%5C+%0A-x%5C+%5Ctextgreater+%5C+-2-3%3B+%5C%5C+%0A-x%5C+%5Ctextgreater+%5C+-5%3B+%5C%5C+%0Ax%5C+%5Ctextless+%5C+5.+%5C%5C+%0A)
Ответ: (-∞;5).
![4) log^2_{2}x+ log_{2}x=0; \\ log_{2}x( log_{2}x+1)=0; \\ log_{2}x=0; \\ x=1; \\ log_{2}x+1=0; \\ log_{2}x=-1; \\ x= \frac{1}{2}. \\](https://tex.z-dn.net/?f=4%29+log%5E2_%7B2%7Dx%2B+log_%7B2%7Dx%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%28+log_%7B2%7Dx%2B1%29%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%3D0%3B+%5C%5C+%0Ax%3D1%3B+%5C%5C+%0A+log_%7B2%7Dx%2B1%3D0%3B+%5C%5C+%0A+log_%7B2%7Dx%3D-1%3B+%5C%5C+%0Ax%3D+%5Cfrac%7B1%7D%7B2%7D.+%5C%5C+++++++++)
ОДЗ:
x>0.
Ответ: 1/2; 1.
5)
![3*4^x+6^x=2*9^x; \\ ](https://tex.z-dn.net/?f=3%2A4%5Ex%2B6%5Ex%3D2%2A9%5Ex%3B+%5C%5C+%0A)
Разделить обе части уравнения на 9^x и перенесем все влево:
![3*( \frac{4}{9})^x+( \frac{6}{9} )^x-2*( \frac{9}{9} )^x=0; \\ 3*( \frac{2}{3} )^{2x}+( \frac{2}{3} )^x-2=0; \\ ( \frac{2}{3} )^x=t; \\ t\ \textgreater \ 0;\\ 3t^2+t-2=0; \\ D=1+24=25; \\ t_{1}= \frac{-1-5}{6}=-1; \\ t_{2}= \frac{-1+5}{6}= \frac{2}{3}; \\ ( \frac{2}{3} )^x= \frac{2}{3}; \\ x=1.](https://tex.z-dn.net/?f=3%2A%28+%5Cfrac%7B4%7D%7B9%7D%29%5Ex%2B%28+%5Cfrac%7B6%7D%7B9%7D+%29%5Ex-2%2A%28+%5Cfrac%7B9%7D%7B9%7D+%29%5Ex%3D0%3B+%5C%5C+%0A3%2A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5E%7B2x%7D%2B%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex-2%3D0%3B+%5C%5C+%0A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex%3Dt%3B++%5C%5C+%0At%5C+%5Ctextgreater+%5C+0%3B%5C%5C+%0A3t%5E2%2Bt-2%3D0%3B+%5C%5C+%0AD%3D1%2B24%3D25%3B+%5C%5C+%0A+t_%7B1%7D%3D+%5Cfrac%7B-1-5%7D%7B6%7D%3D-1%3B+%5C%5C+%0A+t_%7B2%7D%3D+%5Cfrac%7B-1%2B5%7D%7B6%7D%3D+%5Cfrac%7B2%7D%7B3%7D%3B+%5C%5C+%0A%28+%5Cfrac%7B2%7D%7B3%7D+%29%5Ex%3D+%5Cfrac%7B2%7D%7B3%7D%3B+%5C%5C+%0Ax%3D1.+++++++)
Ответ: 1.
(7sin α - 2cos α)/(4sin α - cos α) = 2
Делим числитель и знаменатель дроби на cos α ≠ 0
(7tg α - 2)/(4tg α - 1) = 2 ОДЗ: tg α ≠ 1/4
Умножаем левую и правую части уравнения на (4tg α - 1)
(7tg α - 2) = 2(4tg α - 1)
7tg α - 2 = 8tg α - 2
8tg α - 7tg α = -2 + 2
tg α = 0
Ответ: tg α = 0
5x² + 3x - 8 = 0
D = b² - 4ac = 9 - 4 × 5 × (-8) = 9 + 160 = 169 = 13²
x1 = ( - 3 + 13) / 10 = 1
x2 = ( - 3 - 13) / 10 = - 1,6
Ответ: x1 = 1, x2 = - 1,6.
(2x + 3)( 3x + 1) = 11x + 30
6x² + 11x + 3 = 11x + 30
6x² + 11x - 11x = 30 - 3
6x² = 27
6x² - 27 = 0
2x² - 9 = 0
2x² = 9
x² = 4,5
x1,2 = +/-√4,5
Ответ: x1,2 = +/-√4,5
x² + 4x - 2 = 0
D = b² - 4ac = 16 - 4 × (-2) = 16 + 8 = 24
x1 = ( - 4 + 2√6) / 2 =-2(2-√6)/2 = - (2 - √6) = - 2 + √6
x2 = ( - 4 - 2√6) / 2 =-2(2 + √6)/2 = -(2 + √6) = - 2 - √6
Ответ: x1 = - 2 + √6, x2 = - 2 - √6.