sqrt{4x^2-6x+2}+sqrt{4x^2-1}=2sqrt{4x^2-4x+1} eching

$$\sqrt { 4 x ^ { 2 } - 6 x + 2 } + \sqrt { 4 x ^ { 2 } - 1 } = 2 \sqrt { 4 x ^ { 2 } - 4 x + 1 }$$

x uchun yechish

$$x = \frac{17}{16} = 1\frac{1}{16} = 1,0625$$

$$x=\frac{1}{2}=0,5$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{4x^{2}-6x+2}\right)^{2}+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+\left(\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4x^{2}-6x+2} ga hisoblang va 4x^{2}-6x+2 ni qiymatni oling.

$$4x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+\left(\sqrt{4x^{2}-1}\right)^{2}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4x^{2}-1} ga hisoblang va 4x^{2}-1 ni qiymatni oling.

$$4x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}+4x^{2}-1=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

8x^{2} ni olish uchun 4x^{2} va 4x^{2} ni birlashtirish.

$$8x^{2}-6x+2+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}-1=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

1 olish uchun 2 dan 1 ni ayirish.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=\left(2\sqrt{4x^{2}-4x+1}\right)^{2}$$

\left(2\sqrt{4x^{2}-4x+1}\right)^{2} ni kengaytirish.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=2^{2}\left(\sqrt{4x^{2}-4x+1}\right)^{2}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=4\left(\sqrt{4x^{2}-4x+1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4x^{2}-4x+1} ga hisoblang va 4x^{2}-4x+1 ni qiymatni oling.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=4\left(4x^{2}-4x+1\right)$$

4 ga 4x^{2}-4x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}-6x+1+2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4$$

Tenglamaning ikkala tarafidan 8x^{2}-6x+1 ni ayirish.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4-\left(8x^{2}-6x+1\right)$$

8x^{2}-6x+1 teskarisini topish uchun har birining teskarisini toping.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=16x^{2}-16x+4-8x^{2}+6x-1$$

8x^{2} ni olish uchun 16x^{2} va -8x^{2} ni birlashtirish.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-16x+4+6x-1$$

-10x ni olish uchun -16x va 6x ni birlashtirish.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-10x+4-1$$

3 olish uchun 4 dan 1 ni ayirish.

$$2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}=8x^{2}-10x+3$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

\left(2\sqrt{4x^{2}-6x+2}\sqrt{4x^{2}-1}\right)^{2} ni kengaytirish.

$$2^{2}\left(\sqrt{4x^{2}-6x+2}\right)^{2}\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\left(\sqrt{4x^{2}-6x+2}\right)^{2}\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4x^{2}-6x+2} ga hisoblang va 4x^{2}-6x+2 ni qiymatni oling.

$$4\left(4x^{2}-6x+2\right)\left(\sqrt{4x^{2}-1}\right)^{2}=\left(8x^{2}-10x+3\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4x^{2}-1} ga hisoblang va 4x^{2}-1 ni qiymatni oling.

$$4\left(4x^{2}-6x+2\right)\left(4x^{2}-1\right)=\left(8x^{2}-10x+3\right)^{2}$$

4 ga 4x^{2}-6x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(16x^{2}-24x+8\right)\left(4x^{2}-1\right)=\left(8x^{2}-10x+3\right)^{2}$$

16x^{2}-24x+8 ga 4x^{2}-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$64x^{4}+16x^{2}-96x^{3}+24x-8=\left(8x^{2}-10x+3\right)^{2}$$

8x^{2}-10x+3 kvadratini chiqarish.

$$64x^{4}+16x^{2}-96x^{3}+24x-8=64x^{4}-160x^{3}+148x^{2}-60x+9$$

Ikkala tarafdan 64x^{4} ni ayirish.

$$64x^{4}+16x^{2}-96x^{3}+24x-8-64x^{4}=-160x^{3}+148x^{2}-60x+9$$

0 ni olish uchun 64x^{4} va -64x^{4} ni birlashtirish.

$$16x^{2}-96x^{3}+24x-8=-160x^{3}+148x^{2}-60x+9$$

160x^{3} ni ikki tarafga qo’shing.

$$16x^{2}-96x^{3}+24x-8+160x^{3}=148x^{2}-60x+9$$

64x^{3} ni olish uchun -96x^{3} va 160x^{3} ni birlashtirish.

$$16x^{2}+64x^{3}+24x-8=148x^{2}-60x+9$$

Ikkala tarafdan 148x^{2} ni ayirish.

$$16x^{2}+64x^{3}+24x-8-148x^{2}=-60x+9$$

-132x^{2} ni olish uchun 16x^{2} va -148x^{2} ni birlashtirish.

$$-132x^{2}+64x^{3}+24x-8=-60x+9$$

60x ni ikki tarafga qo’shing.

$$-132x^{2}+64x^{3}+24x-8+60x=9$$

84x ni olish uchun 24x va 60x ni birlashtirish.

$$-132x^{2}+64x^{3}+84x-8=9$$

Ikkala tarafdan 9 ni ayirish.

$$-132x^{2}+64x^{3}+84x-8-9=0$$

-17 olish uchun -8 dan 9 ni ayirish.

$$-132x^{2}+64x^{3}+84x-17=0$$

Tenglamani standart shaklga keltirish uchun uni qayta tartiblash. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$64x^{3}-132x^{2}+84x-17=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart -17 bilan boʻlinadi va q yetakchi koeffisientni 64 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{17}{64},±\frac{17}{32},±\frac{17}{16},±\frac{17}{8},±\frac{17}{4},±\frac{17}{2},±17,±\frac{1}{64},±\frac{1}{32},±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

Eng kichigidan boshlab, mutlaq qiymatgacha butun son qiymatlarni sinab koʻrish orqali ana shunday bitta ildizni toping. Agar butun sonlar ildizlari topilmasa, kasrlarni sinab koʻring.

$$x=\frac{1}{2}$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. 32x^{2}-50x+17 ni olish uchun 64x^{3}-132x^{2}+84x-17 ni 2\left(x-\frac{1}{2}\right)=2x-1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$32x^{2}-50x+17=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 32 ni, b uchun -50 ni va c uchun 17 ni ayiring.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 32\times 17}}{2\times 32}$$

Hisoblarni amalga oshiring.

$$x=\frac{50±18}{64}$$

32x^{2}-50x+17=0 tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=\frac{1}{2} x=\frac{17}{16}$$

Barcha topilgan yechimlar roʻyxati.

$$x=\frac{1}{2} x=\frac{17}{16}$$

\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1} tenglamasida x uchun \frac{1}{2} ni almashtiring.

$$\sqrt{4\times \left(\frac{1}{2}\right)^{2}-6\times \frac{1}{2}+2}+\sqrt{4\times \left(\frac{1}{2}\right)^{2}-1}=2\sqrt{4\times \left(\frac{1}{2}\right)^{2}-4\times \frac{1}{2}+1}$$

Qisqartirish. x=\frac{1}{2} tenglamani qoniqtiradi.

$$0=0$$

\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1} tenglamasida x uchun \frac{17}{16} ni almashtiring.

$$\sqrt{4\times \left(\frac{17}{16}\right)^{2}-6\times \frac{17}{16}+2}+\sqrt{4\times \left(\frac{17}{16}\right)^{2}-1}=2\sqrt{4\times \left(\frac{17}{16}\right)^{2}-4\times \frac{17}{16}+1}$$

Qisqartirish. x=\frac{17}{16} tenglamani qoniqtiradi.

$$\frac{9}{4}=\frac{9}{4}$$

\sqrt{4x^{2}-6x+2}+\sqrt{4x^{2}-1}=2\sqrt{4x^{2}-4x+1} boʻyicha barcha yechimlar roʻyxati.

$$x=\frac{1}{2} x=\frac{17}{16}$$