sqrt{1/2+1/4+1/8+1/16+frac{1}{2}x}=x eching

$$\sqrt{ \frac{ 1 }{ 2 } + \frac{ 1 }{ 4 } + \frac{ 1 }{ 8 } + \frac{ 1 }{ 16 } + \frac{ 1 }{ 2 } x } = x$$

x uchun yechish

$$x = \frac{5}{4} = 1\frac{1}{4} = 1,25$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

2 va 4 ning eng kichik umumiy karralisi 4 ga teng. \frac{1}{2} va \frac{1}{4} ni 4 maxraj bilan kasrlarga aylantirib oling.

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

\frac{2}{4} va \frac{1}{4} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

3 olish uchun 2 va 1'ni qo'shing.

$$\left(\sqrt{\frac{3}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

4 va 8 ning eng kichik umumiy karralisi 8 ga teng. \frac{3}{4} va \frac{1}{8} ni 8 maxraj bilan kasrlarga aylantirib oling.

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

\frac{6}{8} va \frac{1}{8} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

7 olish uchun 6 va 1'ni qo'shing.

$$\left(\sqrt{\frac{7}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

8 va 16 ning eng kichik umumiy karralisi 16 ga teng. \frac{7}{8} va \frac{1}{16} ni 16 maxraj bilan kasrlarga aylantirib oling.

$$\left(\sqrt{\frac{14}{16}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

\frac{14}{16} va \frac{1}{16} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\left(\sqrt{\frac{14+1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

15 olish uchun 14 va 1'ni qo'shing.

$$\left(\sqrt{\frac{15}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

2 daraja ko‘rsatkichini \sqrt{\frac{15}{16}+\frac{1}{2}x} ga hisoblang va \frac{15}{16}+\frac{1}{2}x ni qiymatni oling.

$$\frac{15}{16}+\frac{1}{2}x=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$\frac{15}{16}+\frac{1}{2}x-x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-x^{2}+\frac{1}{2}x+\frac{15}{16}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, \frac{1}{2} ni b va \frac{15}{16} ni c bilan almashtiring.

$$x=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\left(-1\right)\times \frac{15}{16}}}{2\left(-1\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\left(-1\right)\times \frac{15}{16}}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+4\times \frac{15}{16}}}{2\left(-1\right)}$$

4 ni \frac{15}{16} marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1+15}{4}}}{2\left(-1\right)}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{4} ni \frac{15}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\frac{1}{2}±\sqrt{4}}{2\left(-1\right)}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{1}{2}±2}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±2}{-2}$$

x=\frac{-\frac{1}{2}±2}{-2} tenglamasini yeching, bunda ± musbat. -\frac{1}{2} ni 2 ga qo'shish.

$$x=\frac{\frac{3}{2}}{-2}$$

\frac{3}{2} ni -2 ga bo'lish.

$$x=-\frac{3}{4}$$

x=\frac{-\frac{1}{2}±2}{-2} tenglamasini yeching, bunda ± manfiy. -\frac{1}{2} dan 2 ni ayirish.

$$x=-\frac{\frac{5}{2}}{-2}$$

-\frac{5}{2} ni -2 ga bo'lish.

$$x=\frac{5}{4}$$

Tenglama yechildi.

$$x=-\frac{3}{4} x=\frac{5}{4}$$

\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}=x tenglamasida x uchun -\frac{3}{4} ni almashtiring.

$$\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}\left(-\frac{3}{4}\right)}=-\frac{3}{4}$$

Qisqartirish. x=-\frac{3}{4} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$\frac{3}{4}=-\frac{3}{4}$$

\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}=x tenglamasida x uchun \frac{5}{4} ni almashtiring.

$$\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}\times \frac{5}{4}}=\frac{5}{4}$$

Qisqartirish. x=\frac{5}{4} tenglamani qoniqtiradi.

$$\frac{5}{4}=\frac{5}{4}$$

\sqrt{\frac{x}{2}+\frac{15}{16}}=x tenglamasi noyob yechimga ega.

$$x=\frac{5}{4}$$