Упр.735 ГДЗ Колягин Ткачёва 8 класс (Алгебра)

1) ((a+1)/v(a-1)-v(a+1)) :(1/v(a+1)-1/v(a-1));
2) (1/(va+v(a+1))-1/(va-v(a+1))) :v(a+1)/v(a-1);
3) (1/v(1-a)-v(1+a)) :(1/v(1-a^2 )-1);
4) ((a+1)/va+1/(a-va)-a/(1+va))•(v3-av3)/(a+1).
Решить уравнение:
1) 2x^2+x-3=0;
2) 20+8x-x^2=0;
3) 2x^2-9x=35;
4) (x+5)(x-3)=2x-7;
5) 2(x-2)(x+2)=(x+1,5)^2+4(x-5 1/16);
6) (x-3)(x-2)=7x-1.

1. $$\left(\frac{a+1}{\sqrt{a-1}}-\sqrt{a+1}\right):\left(\frac{1}{\sqrt{a+1}}-\frac{1}{\sqrt{a-1}}\right)$$

   $$\frac{a+1-\sqrt{(a+1)(a-1)}}{\sqrt{a-1}}:\frac{\sqrt{a-1}-\sqrt{a+1}}{\sqrt{(a+1)(a-1)}}$$

   $$=\frac{a+1-\sqrt{a^2-1}}{\sqrt{a-1}}\cdot\frac{\sqrt{(a+1)(a-1)}}{\sqrt{a-1}-\sqrt{a+1}}$$

   $$=\frac{\sqrt{a+1}\,(\sqrt{a+1}-\sqrt{a-1})}{1}\cdot\frac{\sqrt{a+1}}{\sqrt{a-1}-\sqrt{a+1}}$$

   $$=-(a+1)=-a-1.$$

2. $$\left(\frac{1}{\sqrt{a}+\sqrt{a+1}}-\frac{1}{\sqrt{a}-\sqrt{a+1}}\right):\frac{\sqrt{a+1}}{\sqrt{a}-1}$$

   $$\frac{\sqrt{a}-\sqrt{a+1}-(\sqrt{a}+\sqrt{a+1})}{(\sqrt{a}-\sqrt{a+1})(\sqrt{a}+\sqrt{a+1})}\cdot\frac{\sqrt{a}-1}{\sqrt{a+1}}$$

   $$=\frac{-2\sqrt{a+1}}{a-(a+1)}\cdot\frac{\sqrt{a}-1}{\sqrt{a+1}}=2\sqrt{a}-1.$$

3. $$\left(\frac{1}{\sqrt{1-a}}-\sqrt{1+a}\right):\left(\frac{1}{\sqrt{1-a^2}}-1\right)$$

   $$\frac{1-\sqrt{(1+a)(1-a)}}{\sqrt{1-a}}:\frac{1-\sqrt{1-a^2}}{\sqrt{1-a^2}}$$

   $$=\frac{1-\sqrt{1-a^2}}{\sqrt{1-a}}:\frac{1-\sqrt{1-a^2}}{\sqrt{1-a^2}}=\sqrt{1+a}.$$

4. $$\left(\frac{a+1}{\sqrt{a}}+\frac{1}{a-\sqrt{a}}-\frac{a}{1+\sqrt{a}}\right)\cdot\frac{\sqrt{3}-a\sqrt{3}}{a+1}$$

   $$\left(\frac{a+1}{\sqrt{a}}+\frac{1}{\sqrt{a}(\sqrt{a}-1)}-\frac{a}{\sqrt{a}+1}\right)\cdot\frac{\sqrt{3}(1-a)}{a+1}$$

   $$=-\sqrt{3}.$$

2. $$2x^2+x-3=0$$

   $$D=1+4\cdot2\cdot3=25$$

   $$x_{1,2}=\frac{-1\pm5}{4}$$

   $$x_1=-\frac{3}{2},\quad x_2=1.$$

3. $$20+8x-x^2=0$$

   $$x^2-8x-20=0$$

   $$x_1+x_2=8,\quad x_1x_2=-20$$

   $$x_1=10,\quad x_2=-2.$$

4. $$2x^2-9x=35$$

   $$2x^2-9x-35=0$$

   $$D=81+4\cdot2\cdot35=361$$

   $$x_{1,2}=\frac{9\pm19}{4}$$

   $$x_1=-\frac{5}{2},\quad x_2=7.$$

5. $$(x+5)(x-3)=2x-7$$

   $$x^2-3x+5x-15=2x-7$$

   $$x^2-8=0$$

   $$x=\pm\sqrt{8}=\pm2\sqrt{2}.$$

6. $$2(x-2)(x+2)=(x+1{,}5)^2+4\left(x-5\frac{1}{16}\right)$$

   $$2(x^2-4)=x^2+3x+2{,}25+4x-\frac{81}{4}$$

   $$2x^2-8=x^2+7x-18$$

   $$x^2-7x+10=0$$

   $$x_1=2,\quad x_2=5.$$

7. $$(x-3)(x-2)=7x-1$$

   $$x^2-5x+6=7x-1$$

   $$x^2-12x+7=0$$

   $$D=144-28=116=4\cdot29$$

   $$x_{1,2}=\frac{12\pm2\sqrt{29}}{2}=6\pm\sqrt{29}.$$

Ответ

1) $$-a-1$$; $$2\sqrt{a}-1$$; $$\sqrt{1+a}$$; $$-\sqrt{3}$$.
2) $$x=-\frac{3}{2},\ 1$$; $$x=-2,\ 10$$; $$x=-\frac{5}{2},\ 7$$; $$x=\pm2\sqrt{2}$$; $$x=2,\ 5$$; $$x=6\pm\sqrt{29}$$.