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Ответил paradiseva
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Задание 1.
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![frac{y^2-11y-26}{9y+18}\
y^2-11y-26=0\
D=b^2-4ac=(-11)^2-4cdot1cdot(-26) =121+104=225=15^2\
y_1= frac{11+15}{2}= 13\
y_2= frac{11-15}{2}= -2\
y^2-11y-26=(y-13)(y+2)\
frac{y^2-11y-26}{9y+18}= frac{(y-13)(y+2)}{9(y+2)}= frac{y-13}{9} \
y=-5 rightarrow frac{-5-13}{9}= frac{-18}{9}=-2\
y=31 rightarrow frac{31-13}{9}= frac{18}{9}=2\
y=112 rightarrow frac{112-13}{9}= frac{99}{9}=11\](https://tex.z-dn.net/?f=+frac%7By%5E2-11y-26%7D%7B9y%2B18%7D%5C%0Ay%5E2-11y-26%3D0%5C%0AD%3Db%5E2-4ac%3D%28-11%29%5E2-4cdot1cdot%28-26%29+%3D121%2B104%3D225%3D15%5E2%5C%0Ay_1%3D+frac%7B11%2B15%7D%7B2%7D%3D+13%5C%0Ay_2%3D+frac%7B11-15%7D%7B2%7D%3D+-2%5C%0Ay%5E2-11y-26%3D%28y-13%29%28y%2B2%29%5C%0A+frac%7By%5E2-11y-26%7D%7B9y%2B18%7D%3D+frac%7B%28y-13%29%28y%2B2%29%7D%7B9%28y%2B2%29%7D%3D+frac%7By-13%7D%7B9%7D++%5C%0Ay%3D-5+rightarrow++frac%7B-5-13%7D%7B9%7D%3D+frac%7B-18%7D%7B9%7D%3D-2%5C%0Ay%3D31+rightarrow++frac%7B31-13%7D%7B9%7D%3D+frac%7B18%7D%7B9%7D%3D2%5C%0Ay%3D112+rightarrow++frac%7B112-13%7D%7B9%7D%3D+frac%7B99%7D%7B9%7D%3D11%5C%0A "frac{y^2-11y-26}{9y+18}\
y^2-11y-26=0\
D=b^2-4ac=(-11)^2-4cdot1cdot(-26) =121+104=225=15^2\
y_1= frac{11+15}{2}= 13\
y_2= frac{11-15}{2}= -2\
y^2-11y-26=(y-13)(y+2)\
frac{y^2-11y-26}{9y+18}= frac{(y-13)(y+2)}{9(y+2)}= frac{y-13}{9} \
y=-5 rightarrow frac{-5-13}{9}= frac{-18}{9}=-2\
y=31 rightarrow frac{31-13}{9}= frac{18}{9}=2\
y=112 rightarrow frac{112-13}{9}= frac{99}{9}=11\")
============== 2 ==============
![frac{x^2-18x+80}{5x-50} \
x^2-18x+80=0\
D=b^2-4ac=(-18)^2-4cdot1cdot80=324-320=4=2^2\
x_1= frac{18+2}{2} =10\
x_2= frac{18-2}{2} =8\
x^2-18x+80=(x-10)(x-8)\
frac{x^2-18x+80}{5x-50}= frac{(x-10)(x-8)}{5(x-10)}= frac{x-8}{5} \
x=-12 rightarrow frac{-12-8}{5}= frac{-20}{5}= -4\
x=8.5 rightarrow frac{8.5-8}{5}= frac{0.5}{5}= 0.1\
x=48 rightarrow frac{48-8}{5}= frac{40}{5}= 8\](https://tex.z-dn.net/?f=+frac%7Bx%5E2-18x%2B80%7D%7B5x-50%7D+%5C%0Ax%5E2-18x%2B80%3D0%5C%0AD%3Db%5E2-4ac%3D%28-18%29%5E2-4cdot1cdot80%3D324-320%3D4%3D2%5E2%5C%0Ax_1%3D+frac%7B18%2B2%7D%7B2%7D+%3D10%5C%0Ax_2%3D+frac%7B18-2%7D%7B2%7D+%3D8%5C%0Ax%5E2-18x%2B80%3D%28x-10%29%28x-8%29%5C%0A+frac%7Bx%5E2-18x%2B80%7D%7B5x-50%7D%3D+frac%7B%28x-10%29%28x-8%29%7D%7B5%28x-10%29%7D%3D+frac%7Bx-8%7D%7B5%7D++%5C%0Ax%3D-12+rightarrow++frac%7B-12-8%7D%7B5%7D%3D+frac%7B-20%7D%7B5%7D%3D+-4%5C%0Ax%3D8.5+rightarrow++frac%7B8.5-8%7D%7B5%7D%3D+frac%7B0.5%7D%7B5%7D%3D+0.1%5C%0Ax%3D48+rightarrow++frac%7B48-8%7D%7B5%7D%3D+frac%7B40%7D%7B5%7D%3D+8%5C "frac{x^2-18x+80}{5x-50} \
x^2-18x+80=0\
D=b^2-4ac=(-18)^2-4cdot1cdot80=324-320=4=2^2\
x_1= frac{18+2}{2} =10\
x_2= frac{18-2}{2} =8\
x^2-18x+80=(x-10)(x-8)\
frac{x^2-18x+80}{5x-50}= frac{(x-10)(x-8)}{5(x-10)}= frac{x-8}{5} \
x=-12 rightarrow frac{-12-8}{5}= frac{-20}{5}= -4\
x=8.5 rightarrow frac{8.5-8}{5}= frac{0.5}{5}= 0.1\
x=48 rightarrow frac{48-8}{5}= frac{40}{5}= 8\")
============== 3 ==============
![frac{x^2-8x-33}{10x+30} \
x^2-8x-33 = 0\
D=b^2-4ac=(-8)^2-4cdot1cdot(-33)=64+132=196=14^2\
x_1= frac{8+14}{2} =11\
x_2= frac{8-14}{2} =-3\
x^2-8x-33 =(x-11)(x+3)\
frac{x^2-8x-33}{10x+30} = frac{(x-11)(x+3)}{10(x+3)} = frac{x-11}{10}\
x=-9 rightarrow frac{-9-11}{10}= frac{-20}{5}= -4\
x=12 rightarrow frac{12-11}{10}= frac{1}{10}\
x=111 rightarrow frac{111-11}{10}= frac{100}{5}= 20\](https://tex.z-dn.net/?f=+frac%7Bx%5E2-8x-33%7D%7B10x%2B30%7D+%5C%0Ax%5E2-8x-33+%3D+0%5C%0AD%3Db%5E2-4ac%3D%28-8%29%5E2-4cdot1cdot%28-33%29%3D64%2B132%3D196%3D14%5E2%5C%0Ax_1%3D+frac%7B8%2B14%7D%7B2%7D+%3D11%5C%0Ax_2%3D+frac%7B8-14%7D%7B2%7D+%3D-3%5C%0Ax%5E2-8x-33+%3D%28x-11%29%28x%2B3%29%5C%0A+frac%7Bx%5E2-8x-33%7D%7B10x%2B30%7D+%3D+frac%7B%28x-11%29%28x%2B3%29%7D%7B10%28x%2B3%29%7D+%3D+frac%7Bx-11%7D%7B10%7D%5C%0A+x%3D-9+rightarrow+frac%7B-9-11%7D%7B10%7D%3D+frac%7B-20%7D%7B5%7D%3D+-4%5C%0A+x%3D12+rightarrow+frac%7B12-11%7D%7B10%7D%3D+frac%7B1%7D%7B10%7D%5C%0A+x%3D111+rightarrow+frac%7B111-11%7D%7B10%7D%3D+frac%7B100%7D%7B5%7D%3D+20%5C%0A "frac{x^2-8x-33}{10x+30} \
x^2-8x-33 = 0\
D=b^2-4ac=(-8)^2-4cdot1cdot(-33)=64+132=196=14^2\
x_1= frac{8+14}{2} =11\
x_2= frac{8-14}{2} =-3\
x^2-8x-33 =(x-11)(x+3)\
frac{x^2-8x-33}{10x+30} = frac{(x-11)(x+3)}{10(x+3)} = frac{x-11}{10}\
x=-9 rightarrow frac{-9-11}{10}= frac{-20}{5}= -4\
x=12 rightarrow frac{12-11}{10}= frac{1}{10}\
x=111 rightarrow frac{111-11}{10}= frac{100}{5}= 20\")
============== 4 ==============
![frac{8y-56}{y^2-27y+140}\
y^2-27y+140=0\
D=b^2-4ac=(-27)^2-4cdot1cdot140=729-560=169=13^2\
y_1= frac{27+13}{2}= frac{40}{2}=20\
y_2= frac{27-13}{2}= frac{14}{2}=7\
y^2-27y+140= (y-20)(y-7)\
frac{8y-56}{y^2-27y+140}= frac{8(y-7)}{(y-20)(y-7)}= frac{8}{y-20}\
y = -4 rightarrow frac{8}{-4-20}= frac{8}{-24}= - frac{1}{3}\
y = 22.5 rightarrow frac{8}{22.5-20}= frac{8}{2.5}= 3.2\
y = 24 rightarrow frac{8}{24-20}= frac{8}{4}= 2\](https://tex.z-dn.net/?f=+frac%7B8y-56%7D%7By%5E2-27y%2B140%7D%5C%0A+y%5E2-27y%2B140%3D0%5C%0AD%3Db%5E2-4ac%3D%28-27%29%5E2-4cdot1cdot140%3D729-560%3D169%3D13%5E2%5C%0Ay_1%3D+frac%7B27%2B13%7D%7B2%7D%3D+frac%7B40%7D%7B2%7D%3D20%5C++%0Ay_2%3D+frac%7B27-13%7D%7B2%7D%3D+frac%7B14%7D%7B2%7D%3D7%5C++%0A+y%5E2-27y%2B140%3D+%28y-20%29%28y-7%29%5C%0A+frac%7B8y-56%7D%7By%5E2-27y%2B140%7D%3D+frac%7B8%28y-7%29%7D%7B%28y-20%29%28y-7%29%7D%3D+frac%7B8%7D%7By-20%7D%5C%0A+y+%3D+-4+rightarrow+frac%7B8%7D%7B-4-20%7D%3D+frac%7B8%7D%7B-24%7D%3D+-+frac%7B1%7D%7B3%7D%5C+%0A+y+%3D+22.5+rightarrow+frac%7B8%7D%7B22.5-20%7D%3D+frac%7B8%7D%7B2.5%7D%3D+3.2%5C+%0A+y+%3D+24+rightarrow+frac%7B8%7D%7B24-20%7D%3D+frac%7B8%7D%7B4%7D%3D+2%5C+ "frac{8y-56}{y^2-27y+140}\
y^2-27y+140=0\
D=b^2-4ac=(-27)^2-4cdot1cdot140=729-560=169=13^2\
y_1= frac{27+13}{2}= frac{40}{2}=20\
y_2= frac{27-13}{2}= frac{14}{2}=7\
y^2-27y+140= (y-20)(y-7)\
frac{8y-56}{y^2-27y+140}= frac{8(y-7)}{(y-20)(y-7)}= frac{8}{y-20}\
y = -4 rightarrow frac{8}{-4-20}= frac{8}{-24}= - frac{1}{3}\
y = 22.5 rightarrow frac{8}{22.5-20}= frac{8}{2.5}= 3.2\
y = 24 rightarrow frac{8}{24-20}= frac{8}{4}= 2\")
Задание 2.
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![(1+ sqrt{ frac{a-x}{a+x} } )(1- sqrt{ frac{a-x}{a+x} } )=1^2- (sqrt{ frac{a-x}{a+x} })^2=1- frac{a-x}{a+x} }= frac{a+x-a+x}{a+x} =\
= frac{2x}{a+x} \
a=5, x=4 rightarrow frac{2x}{a+x} = frac{2cdot4}{5+4}= frac{8}{9}](https://tex.z-dn.net/?f=%281%2B+sqrt%7B+frac%7Ba-x%7D%7Ba%2Bx%7D+%7D+%29%281-+sqrt%7B+frac%7Ba-x%7D%7Ba%2Bx%7D+%7D+%29%3D1%5E2-+%28sqrt%7B+frac%7Ba-x%7D%7Ba%2Bx%7D+%7D%29%5E2%3D1-+frac%7Ba-x%7D%7Ba%2Bx%7D+%7D%3D+frac%7Ba%2Bx-a%2Bx%7D%7Ba%2Bx%7D+%3D%5C%0A%3D+frac%7B2x%7D%7Ba%2Bx%7D+%5C%0Aa%3D5%2C+x%3D4+rightarrow++frac%7B2x%7D%7Ba%2Bx%7D+%3D+frac%7B2cdot4%7D%7B5%2B4%7D%3D+frac%7B8%7D%7B9%7D+ "(1+ sqrt{ frac{a-x}{a+x} } )(1- sqrt{ frac{a-x}{a+x} } )=1^2- (sqrt{ frac{a-x}{a+x} })^2=1- frac{a-x}{a+x} }= frac{a+x-a+x}{a+x} =\
= frac{2x}{a+x} \
a=5, x=4 rightarrow frac{2x}{a+x} = frac{2cdot4}{5+4}= frac{8}{9}")
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Задание 2.
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