пусть x литров- пропускает 1 труба,тогда х+4 л пропускает 2 труба
48/x=48/(x+4)+2
48x+192=48x+2x^2+8x
2x^2+8x-192=0
x^2+4x-96=9
D=16+384=400
x1=-(4+20)/2=8
x2=(-4-20)/2=-12 не подходит ответ так как отрицательное число
Ответ: 8 литров пропускает 1 труба за 1 минуту
12+0,4×корень из 40×360=12+48=60
Вторая четверть x<=0, y>=0
третья x<=0, y<=0
четвертая x>=0, y<=0
1)
(2x²-x-1+6)² +3(2x²-x-1)-10=0
y=2x²-x-1
(y+6)²+3y-10=0
y²+12y+36+3y-10=0
y²+15y+26=0
D=225-104=121
y₁=(-15-11)/2=-13
y₂=(-15+11)/2=-2
При у=-13
2x²-x-1= -13
2x²-x-1+13=0
2x²-x+12=0
D=1-4*2*12<0
нет решений.
При у=-2
2x²-x-1= -2
2x²-x-1+2=0
2x²-x+1=0
D=1-4*2*1<0
нет решений.
Ответ: нет решений.
2.
![\frac{x^2-2x}{4x-3}+5= \frac{16x-12}{2x-x^2} \\ \\ x \neq \frac{3}{4} \\ x \neq 0 \\ x \neq 2](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D%2B5%3D+%5Cfrac%7B16x-12%7D%7B2x-x%5E2%7D+%5C%5C+%0A+%5C%5C+%0Ax+%5Cneq++%5Cfrac%7B3%7D%7B4%7D+%5C%5C+%0Ax+%5Cneq+0+%5C%5C+%0Ax+%5Cneq+2+++)
![\frac{x^2-2x}{4x-3}+5=- \frac{4(4x-3)}{x^2-2x} \\ \\ \frac{x^2-2x}{4x-3}+4 *\frac{4x-3}{x^2-2x}+5=0](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D%2B5%3D-+%5Cfrac%7B4%284x-3%29%7D%7Bx%5E2-2x%7D+%5C%5C+%0A+%5C%5C+%0A+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D%2B4+%2A%5Cfrac%7B4x-3%7D%7Bx%5E2-2x%7D%2B5%3D0++++)
![y= \frac{x^2-2x}{4x-3} \\ \\ y+ \frac{4}{y}+5=0 \\ y^2+5y+4=0 \\ D=25-16=9 \\ y_{1}= \frac{-5-3}{2}=-4 \\ y \\ _{2}= \frac{-5+3}{2}=-1](https://tex.z-dn.net/?f=y%3D+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D+%5C%5C+%0A+%5C%5C+%0Ay%2B+%5Cfrac%7B4%7D%7By%7D%2B5%3D0+%5C%5C+%0Ay%5E2%2B5y%2B4%3D0+%5C%5C+%0AD%3D25-16%3D9+%5C%5C+%0Ay_%7B1%7D%3D+%5Cfrac%7B-5-3%7D%7B2%7D%3D-4+%5C%5C+%0Ay+%5C%5C+_%7B2%7D%3D+%5Cfrac%7B-5%2B3%7D%7B2%7D%3D-1++++)
При у= -4
![\frac{x^2-2x}{4x-3}= -4 \\ \\ x^2-2x= -4(4x-3) \\ x^2-2x=-16x+12 \\ x^2-2x+16x-12=0 \\ x^2+14x-12=0 \\ D=196+48=244 \\ x_{1}= \frac{-14- \sqrt{244} }{2}= \frac{-14-2 \sqrt{61} }{2}=-7- \sqrt{61} \\ x_{2}=-7+ \sqrt{61}](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D%3D+-4+%5C%5C+%0A+%5C%5C+%0Ax%5E2-2x%3D+-4%284x-3%29++%5C%5C+%0Ax%5E2-2x%3D-16x%2B12+%5C%5C+%0Ax%5E2-2x%2B16x-12%3D0+%5C%5C+%0Ax%5E2%2B14x-12%3D0+%5C%5C+%0AD%3D196%2B48%3D244+%5C%5C+%0Ax_%7B1%7D%3D+%5Cfrac%7B-14-+%5Csqrt%7B244%7D+%7D%7B2%7D%3D+%5Cfrac%7B-14-2+%5Csqrt%7B61%7D+%7D%7B2%7D%3D-7-+%5Csqrt%7B61%7D++++%5C%5C+%0Ax_%7B2%7D%3D-7%2B+%5Csqrt%7B61%7D+)
При у= -1
![\frac{x^2-2x}{4x-3}=-1 \\ \\ x^2-2x=-1(4x-3) \\ x^2-2x=-4x+3 \\ x^2-2x+4x-3=0 \\ x^2+2x-3=0 \\ D=4+12=16 \\ x_{1}= \frac{-2-4}{2}=-3 \\ x_{2}= \frac{-2+4}{2}=1](https://tex.z-dn.net/?f=+%5Cfrac%7Bx%5E2-2x%7D%7B4x-3%7D%3D-1+%5C%5C+%0A+%5C%5C+%0Ax%5E2-2x%3D-1%284x-3%29+%5C%5C+%0Ax%5E2-2x%3D-4x%2B3+%5C%5C+%0Ax%5E2-2x%2B4x-3%3D0+%5C%5C+%0Ax%5E2%2B2x-3%3D0+%5C%5C+%0AD%3D4%2B12%3D16+%5C%5C+%0Ax_%7B1%7D%3D+%5Cfrac%7B-2-4%7D%7B2%7D%3D-3+%5C%5C+%0Ax_%7B2%7D%3D+%5Cfrac%7B-2%2B4%7D%7B2%7D%3D1+++)
Ответ: -7-√61; -3; -7+√61; 1.
3.
![(\frac{4x-5}{3x+2})^2+( \frac{3x+2}{5-4x} )^2=4.25 \\ \\ ( \frac{4x-5}{3x+2} )^2+(- \frac{3x+2}{4x-5} )^2=4.25 \\ \\ (\frac{4x-5}{3x+2})^2 +( \frac{3x+2}{4x-5} )^2=4.25 \\ \\ x \neq - \frac{2}{3} \\ x \neq \frac{5}{4}](https://tex.z-dn.net/?f=+%28%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%29%5E2%2B%28+%5Cfrac%7B3x%2B2%7D%7B5-4x%7D+%29%5E2%3D4.25+%5C%5C+%0A+%5C%5C+%0A%28+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D+%29%5E2%2B%28-+%5Cfrac%7B3x%2B2%7D%7B4x-5%7D+%29%5E2%3D4.25+%5C%5C+%0A+%5C%5C+%0A+%28%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%29%5E2+%2B%28+%5Cfrac%7B3x%2B2%7D%7B4x-5%7D+%29%5E2%3D4.25++%5C%5C+%0A+%5C%5C+%0Ax+%5Cneq+-+%5Cfrac%7B2%7D%7B3%7D+%5C%5C+%0Ax+%5Cneq++%5Cfrac%7B5%7D%7B4%7D++)
![y= \frac{4x-5}{3x+2} \\ \\ y^2+( \frac{1}{y} )^2-4.25=0 \\ y^2+ \frac{1}{y^2}-4.25=0 \\ \\ a=y^2](https://tex.z-dn.net/?f=y%3D+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D+%5C%5C+%0A+%5C%5C+%0Ay%5E2%2B%28+%5Cfrac%7B1%7D%7By%7D+%29%5E2-4.25%3D0+%5C%5C+%0Ay%5E2%2B+%5Cfrac%7B1%7D%7By%5E2%7D-4.25%3D0+%5C%5C+%0A+%5C%5C+%0Aa%3Dy%5E2)
![a+ \frac{1}{a}-4.25=0 \\ a^2-4.25a+1=0 \\ D=4.25^2-4=14.0625=3.75^2 \\ a_{1}= \frac{4.25-3.75}{2}=0.25 \\ a_{2}= \frac{4.25+3.75}{2}=4](https://tex.z-dn.net/?f=a%2B+%5Cfrac%7B1%7D%7Ba%7D-4.25%3D0+%5C%5C+%0Aa%5E2-4.25a%2B1%3D0+%5C%5C+%0AD%3D4.25%5E2-4%3D14.0625%3D3.75%5E2+%5C%5C+%0Aa_%7B1%7D%3D+%5Cfrac%7B4.25-3.75%7D%7B2%7D%3D0.25+%5C%5C+%0Aa_%7B2%7D%3D+%5Cfrac%7B4.25%2B3.75%7D%7B2%7D%3D4+++)
При а=0,25
у²=0,25
у₁= 0,5
у₂= -0,5
При а=4
у²=4
у₁=2
у₂= -2
При у=0,5
![\frac{4x-5}{3x+2}=0.5 \\ 4x-5=0.5(3x+2) \\ 4x-5=1.5x+1 \\ 4x-1.5x=1+5 \\ 2.5x=6 \\ x=2.4](https://tex.z-dn.net/?f=+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%3D0.5+%5C%5C+%0A4x-5%3D0.5%283x%2B2%29+%5C%5C+%0A4x-5%3D1.5x%2B1+%5C%5C+%0A4x-1.5x%3D1%2B5+%5C%5C+%0A2.5x%3D6+%5C%5C+%0Ax%3D2.4+)
При у=-0,5
![\frac{4x-5}{3x+2}=-0.5 \\ 4x-5=-0.5(3x+2) \\ 4x-5=-1.5x-1 \\ 4x+1.5x=5-1 \\ 5.5x=4 \\ \frac{11}{2}x=4 \\ x= \frac{8}{11}](https://tex.z-dn.net/?f=+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%3D-0.5+%5C%5C+%0A4x-5%3D-0.5%283x%2B2%29+%5C%5C+%0A4x-5%3D-1.5x-1+%5C%5C+%0A4x%2B1.5x%3D5-1+%5C%5C+%0A5.5x%3D4+%5C%5C+%0A+%5Cfrac%7B11%7D%7B2%7Dx%3D4+%5C%5C+%0Ax%3D+%5Cfrac%7B8%7D%7B11%7D+++)
При у=2
![\frac{4x-5}{3x+2}=2 \\ 4x-5=2(3x+2) \\ 4x-5=6x+4 \\ 4x-6x=4+5 \\ -2x=9 \\ x=-4.5](https://tex.z-dn.net/?f=+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%3D2+%5C%5C+%0A4x-5%3D2%283x%2B2%29+%5C%5C+%0A4x-5%3D6x%2B4+%5C%5C+%0A4x-6x%3D4%2B5+%5C%5C+%0A-2x%3D9+%5C%5C+%0Ax%3D-4.5+)
При у=-2
![\frac{4x-5}{3x+2}=-2 \\ 4x-5=-2(3x+2) \\ 4x-5=-6x-4 \\ 4x+6x=5-4 \\ 10x=1 \\ 0.1](https://tex.z-dn.net/?f=+%5Cfrac%7B4x-5%7D%7B3x%2B2%7D%3D-2+%5C%5C+%0A4x-5%3D-2%283x%2B2%29+%5C%5C+%0A4x-5%3D-6x-4+%5C%5C+%0A4x%2B6x%3D5-4+%5C%5C+%0A10x%3D1+%5C%5C+%0A0.1+)
Ответ: -4,5; 0,1; 8/11; 2,4
1) a_(n+1)=2((n+1)-10)=2n-18
a_n+2=2(n+2-10)=2n-16
a_n+5=2(n+5-10)=2n-10
2) a_n=7*2^(-n-2)
a_n+1=7*2^(-n-1-2)=7*2^(-n-3)
a_n+2=7*2^(-n-4)
a_n+5=7*2^(-n-7)