sqrt{5x+2}-4/sqrt{5x+2}=5 eching

$$\sqrt { 5 x + 2 } - \frac { 4 } { \sqrt { 5 x + 2 } } = 5$$

x uchun yechish

$$x=\frac{\sqrt{41}}{2}+\frac{29}{10}\approx 6,101562119$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -\frac{4}{\sqrt{5x+2}} ni ayirish.

$$\sqrt{5x+2}=5-\left(-\frac{4}{\sqrt{5x+2}}\right)$$

1 hosil qilish uchun -1 va -1 ni ko'paytirish.

$$\sqrt{5x+2}=5+\frac{4}{\sqrt{5x+2}}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 5 ni \frac{\sqrt{5x+2}}{\sqrt{5x+2}} marotabaga ko'paytirish.

$$\sqrt{5x+2}=\frac{5\sqrt{5x+2}}{\sqrt{5x+2}}+\frac{4}{\sqrt{5x+2}}$$

\frac{5\sqrt{5x+2}}{\sqrt{5x+2}} va \frac{4}{\sqrt{5x+2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\sqrt{5x+2}=\frac{5\sqrt{5x+2}+4}{\sqrt{5x+2}}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{5x+2}\right)^{2}=\left(\frac{5\sqrt{5x+2}+4}{\sqrt{5x+2}}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x+2} ga hisoblang va 5x+2 ni qiymatni oling.

$$5x+2=\left(\frac{5\sqrt{5x+2}+4}{\sqrt{5x+2}}\right)^{2}$$

\frac{5\sqrt{5x+2}+4}{\sqrt{5x+2}}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$5x+2=\frac{\left(5\sqrt{5x+2}+4\right)^{2}}{\left(\sqrt{5x+2}\right)^{2}}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(5\sqrt{5x+2}+4\right)^{2} kengaytirilishi uchun ishlating.

$$5x+2=\frac{25\left(\sqrt{5x+2}\right)^{2}+40\sqrt{5x+2}+16}{\left(\sqrt{5x+2}\right)^{2}}$$

2 daraja ko‘rsatkichini \sqrt{5x+2} ga hisoblang va 5x+2 ni qiymatni oling.

$$5x+2=\frac{25\left(5x+2\right)+40\sqrt{5x+2}+16}{\left(\sqrt{5x+2}\right)^{2}}$$

25 ga 5x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+2=\frac{125x+50+40\sqrt{5x+2}+16}{\left(\sqrt{5x+2}\right)^{2}}$$

66 olish uchun 50 va 16'ni qo'shing.

$$5x+2=\frac{125x+66+40\sqrt{5x+2}}{\left(\sqrt{5x+2}\right)^{2}}$$

2 daraja ko‘rsatkichini \sqrt{5x+2} ga hisoblang va 5x+2 ni qiymatni oling.

$$5x+2=\frac{125x+66+40\sqrt{5x+2}}{5x+2}$$

Tenglamaning ikkala tarafini 5x+2 ga ko'paytirish.

$$5x\left(5x+2\right)+\left(5x+2\right)\times 2=125x+66+40\sqrt{5x+2}$$

Tenglamaning ikkala tarafidan 125x+66 ni ayirish.

$$5x\left(5x+2\right)+\left(5x+2\right)\times 2-\left(125x+66\right)=40\sqrt{5x+2}$$

5x ga 5x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$25x^{2}+10x+\left(5x+2\right)\times 2-\left(125x+66\right)=40\sqrt{5x+2}$$

5x+2 ga 2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$25x^{2}+10x+10x+4-\left(125x+66\right)=40\sqrt{5x+2}$$

20x ni olish uchun 10x va 10x ni birlashtirish.

$$25x^{2}+20x+4-\left(125x+66\right)=40\sqrt{5x+2}$$

125x+66 teskarisini topish uchun har birining teskarisini toping.

$$25x^{2}+20x+4-125x-66=40\sqrt{5x+2}$$

-105x ni olish uchun 20x va -125x ni birlashtirish.

$$25x^{2}-105x+4-66=40\sqrt{5x+2}$$

-62 olish uchun 4 dan 66 ni ayirish.

$$25x^{2}-105x-62=40\sqrt{5x+2}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(25x^{2}-105x-62\right)^{2}=\left(40\sqrt{5x+2}\right)^{2}$$

25x^{2}-105x-62 kvadratini chiqarish.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844=\left(40\sqrt{5x+2}\right)^{2}$$

\left(40\sqrt{5x+2}\right)^{2} ni kengaytirish.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844=40^{2}\left(\sqrt{5x+2}\right)^{2}$$

2 daraja ko‘rsatkichini 40 ga hisoblang va 1600 ni qiymatni oling.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844=1600\left(\sqrt{5x+2}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x+2} ga hisoblang va 5x+2 ni qiymatni oling.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844=1600\left(5x+2\right)$$

1600 ga 5x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844=8000x+3200$$

Ikkala tarafdan 8000x ni ayirish.

$$625x^{4}-5250x^{3}+7925x^{2}+13020x+3844-8000x=3200$$

5020x ni olish uchun 13020x va -8000x ni birlashtirish.

$$625x^{4}-5250x^{3}+7925x^{2}+5020x+3844=3200$$

Ikkala tarafdan 3200 ni ayirish.

$$625x^{4}-5250x^{3}+7925x^{2}+5020x+3844-3200=0$$

644 olish uchun 3844 dan 3200 ni ayirish.

$$625x^{4}-5250x^{3}+7925x^{2}+5020x+644=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 644 bilan boʻlinadi va q yetakchi koeffisientni 625 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{644}{625},±\frac{644}{125},±\frac{644}{25},±\frac{644}{5},±644,±\frac{322}{625},±\frac{322}{125},±\frac{322}{25},±\frac{322}{5},±322,±\frac{161}{625},±\frac{161}{125},±\frac{161}{25},±\frac{161}{5},±161,±\frac{92}{625},±\frac{92}{125},±\frac{92}{25},±\frac{92}{5},±92,±\frac{46}{625},±\frac{46}{125},±\frac{46}{25},±\frac{46}{5},±46,±\frac{28}{625},±\frac{28}{125},±\frac{28}{25},±\frac{28}{5},±28,±\frac{23}{625},±\frac{23}{125},±\frac{23}{25},±\frac{23}{5},±23,±\frac{14}{625},±\frac{14}{125},±\frac{14}{25},±\frac{14}{5},±14,±\frac{7}{625},±\frac{7}{125},±\frac{7}{25},±\frac{7}{5},±7,±\frac{4}{625},±\frac{4}{125},±\frac{4}{25},±\frac{4}{5},±4,±\frac{2}{625},±\frac{2}{125},±\frac{2}{25},±\frac{2}{5},±2,±\frac{1}{625},±\frac{1}{125},±\frac{1}{25},±\frac{1}{5},±1$$

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

$$x=-\frac{1}{5}$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. 125x^{3}-1075x^{2}+1800x+644 ni olish uchun 625x^{4}-5250x^{3}+7925x^{2}+5020x+644 ni 5\left(x+\frac{1}{5}\right)=5x+1 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$125x^{3}-1075x^{2}+1800x+644=0$$

Ratsional ildiz teoremasiga koʻra, koʻphadlarning barcha ratsional ildizlari \frac{p}{q} shakli ichida, bu yerda p konstant shart 644 bilan boʻlinadi va q yetakchi koeffisientni 125 boʻladi. Barcha nomzodlarni oching \frac{p}{q}.

$$±\frac{644}{125},±\frac{644}{25},±\frac{644}{5},±644,±\frac{322}{125},±\frac{322}{25},±\frac{322}{5},±322,±\frac{161}{125},±\frac{161}{25},±\frac{161}{5},±161,±\frac{92}{125},±\frac{92}{25},±\frac{92}{5},±92,±\frac{46}{125},±\frac{46}{25},±\frac{46}{5},±46,±\frac{28}{125},±\frac{28}{25},±\frac{28}{5},±28,±\frac{23}{125},±\frac{23}{25},±\frac{23}{5},±23,±\frac{14}{125},±\frac{14}{25},±\frac{14}{5},±14,±\frac{7}{125},±\frac{7}{25},±\frac{7}{5},±7,±\frac{4}{125},±\frac{4}{25},±\frac{4}{5},±4,±\frac{2}{125},±\frac{2}{25},±\frac{2}{5},±2,±\frac{1}{125},±\frac{1}{25},±\frac{1}{5},±1$$

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

$$x=\frac{14}{5}$$

Faktor teoremasiga koʻra, x-k har bir k ildizining faktoridir. 25x^{2}-145x-46 ni olish uchun 125x^{3}-1075x^{2}+1800x+644 ni 5\left(x-\frac{14}{5}\right)=5x-14 ga bo‘ling. Natija 0 ga teng boʻlgandagi tenglamani yeching.

$$25x^{2}-145x-46=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun 25 ni, b uchun -145 ni va c uchun -46 ni ayiring.

$$x=\frac{-\left(-145\right)±\sqrt{\left(-145\right)^{2}-4\times 25\left(-46\right)}}{2\times 25}$$

Hisoblarni amalga oshiring.

$$x=\frac{145±25\sqrt{41}}{50}$$

25x^{2}-145x-46=0 tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

$$x=-\frac{\sqrt{41}}{2}+\frac{29}{10} x=\frac{\sqrt{41}}{2}+\frac{29}{10}$$

Barcha topilgan yechimlar roʻyxati.

$$x=-\frac{1}{5} x=\frac{14}{5} x=-\frac{\sqrt{41}}{2}+\frac{29}{10} x=\frac{\sqrt{41}}{2}+\frac{29}{10}$$

\sqrt{5x+2}-\frac{4}{\sqrt{5x+2}}=5 tenglamasida x uchun -\frac{1}{5} ni almashtiring.

$$\sqrt{5\left(-\frac{1}{5}\right)+2}-\frac{4}{\sqrt{5\left(-\frac{1}{5}\right)+2}}=5$$

Qisqartirish. x=-\frac{1}{5} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$-3=5$$

\sqrt{5x+2}-\frac{4}{\sqrt{5x+2}}=5 tenglamasida x uchun \frac{14}{5} ni almashtiring.

$$\sqrt{5\times \frac{14}{5}+2}-\frac{4}{\sqrt{5\times \frac{14}{5}+2}}=5$$

Qisqartirish. x=\frac{14}{5} qiymati bu tenglamani qoniqtirmaydi.

$$3=5$$

\sqrt{5x+2}-\frac{4}{\sqrt{5x+2}}=5 tenglamasida x uchun -\frac{\sqrt{41}}{2}+\frac{29}{10} ni almashtiring.

$$\sqrt{5\left(-\frac{\sqrt{41}}{2}+\frac{29}{10}\right)+2}-\frac{4}{\sqrt{5\left(-\frac{\sqrt{41}}{2}+\frac{29}{10}\right)+2}}=5$$

Qisqartirish. x=-\frac{\sqrt{41}}{2}+\frac{29}{10} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$-5=5$$

\sqrt{5x+2}-\frac{4}{\sqrt{5x+2}}=5 tenglamasida x uchun \frac{\sqrt{41}}{2}+\frac{29}{10} ni almashtiring.

$$\sqrt{5\left(\frac{\sqrt{41}}{2}+\frac{29}{10}\right)+2}-\frac{4}{\sqrt{5\left(\frac{\sqrt{41}}{2}+\frac{29}{10}\right)+2}}=5$$

Qisqartirish. x=\frac{\sqrt{41}}{2}+\frac{29}{10} tenglamani qoniqtiradi.

$$5=5$$

\sqrt{5x+2}=\frac{5\sqrt{5x+2}+4}{\sqrt{5x+2}} tenglamasi noyob yechimga ega.

$$x=\frac{\sqrt{41}}{2}+\frac{29}{10}$$