Question

50 БАЛЛОВ МАТЕМАТИКА
помогите пожалуйста
Иррациональные уравнения​

Answer 1

1)

$displaystyle tt sqrt{65-x^2}=4\displaystyle tt 65-x^2=16\displaystyle tt -x^2=-49\displaystyle tt x^2=49\displaystyle tt bold{x_1=7}\displaystyle tt bold{x_2=-7}$

2)

$displaystyle tt x-sqrt{x+4}=8\displaystyle tt -sqrt{x+4}=8-x\displaystyle tt sqrt{x+4}=-8+x\displaystyle tt x+4=x^2-16x+64\displaystyle tt -x^2+17x-60=0\displaystyle tt x^2-17x+60=0\displaystyle tt x^2-5x-12x+60=0\displaystyle tt x(x-5)-12(x-5)=0\displaystyle tt (x-5)(x-12)=0\displaystyle tt bold{x_1=5}\displaystyle tt bold{x_2=12}$

3)

$displaystyle tt sqrt{8-x}cdotsqrt{5-x}=2\displaystyle tt sqrt{(8-x)(5-x)}=2\displaystyle tt sqrt{40-8x-5x+x^2}=2\displaystyle tt 40-13x+x^2=4\displaystyle tt x^2-13x+36=0\displaystyle tt x^2-4x-9x+36=0\displaystyle tt x(x-4)-9(x-4)=0\displaystyle tt (x-4)(x-9)=0\displaystyle tt bold{x_1=4}$

в ответ идёт только один корень, так как 9 не подходит

4)

$displaystyle tt sqrt{x^2-8x+16}-2=0\displaystyle tt sqrt{(x-4)^2}-2=0\displaystyle tt |x-4|-2=0\displaystyle tt |x-4|=2\displaystyle tt x-4=2\displaystyle tt bold{x_1=6}\displaystyle tt x-4=-2\displaystyle tt bold{x_2=2}$

5)

$displaystyle tt sqrt{x}cdotsqrt{15+x}=2x\displaystyle tt sqrt{xcdot(15+x)}=2x\displaystyle tt sqrt{15x+x^2}=2x\displaystyle tt 15x+x^2=4x^2\displaystyle tt 15x-3x^2=0\displaystyle tt 3x(5-x)=0\displaystyle tt x(5-x)=0\displaystyle tt bold{x_1=0}\displaystyle tt bold{x_2=5}$

Answer 2

1)

$sqrt{65-x^2}=4\left(sqrt{65-x^2}right)^2=4^2\65-x^2=16\-x^2=-49\x^2=49\x=sqrt{49},:x=-sqrt{49}\x=7; x=-7$

2)

$x-sqrt{x+4}=8\-sqrt{x+4}=8-x\(-sqrt{x+4})^2=(8-x)^2\x+4=64-16x+x^2\x^2-17x+60=0\x_{1}=frac{-left(-17right)+sqrt{left(-17right)^2-4cdot :1cdot :60}}{2cdot :1}=frac{17+sqrt{49}}{2}=12\x_{2}=frac{-left(-17right)-sqrt{left(-17right)^2-4cdot :1cdot :60}}{2cdot :1}=frac{17-sqrt{49}}{2}=5$

3)

$sqrt{8-x}cdot sqrt{5-x}=2\(sqrt{8-x}cdot sqrt{5-x})^2=2^2\(8-x)(5-x)=4\40-13x+x^2=4\x^2-13x+36=0\x_{1}=frac{-left(-13right)+sqrt{left(-13right)^2-4cdot :1cdot :36}}{2cdot :1}=frac{13+sqrt{25}}{2}=9\x_{2}=frac{-left(-13right)-sqrt{left(-13right)^2-4cdot :1cdot :36}}{2cdot :1}=frac{13-sqrt{25}}{2}=4$

x = 9 не подходит, в ответ не заносить

4)

$sqrt{x^2-8x+16}-2=0\left(sqrt{x^2-8x+16}right)^2=2^2\x^2-8x+16=4\x^2-8x+12=0\x_{1}=frac{-left(-8right)+sqrt{left(-8right)^2-4cdot :1cdot :12}}{2cdot :1}=frac{8+sqrt{16}}{2}=6\x_{2}=frac{-left(-8right)-sqrt{left(-8right)^2-4cdot :1cdot :12}}{2cdot :1}=frac{8-sqrt{16}}{2}=2$

5)

$sqrt{x}cdot sqrt{15+x}=2x\left(sqrt{x}sqrt{15+x}right)^2=left(2xright)^2\x(15+x)=4x^2\15x+x^2=4x^2\-3x^2+15x=0\-3x(x-5)=0\ x=0; x=5$