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1+\frac{1}{1+\frac{1}{\frac{x}{x}+\frac{1}{x}}}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x}{x} marotabaga ko'paytirish.
1+\frac{1}{1+\frac{1}{\frac{x+1}{x}}}=x
\frac{x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
1+\frac{1}{1+\frac{x}{x+1}}=x
x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{x+1}{x} ga bo'lish 1 ga k'paytirish \frac{x+1}{x} ga qaytarish.
1+\frac{1}{\frac{x+1}{x+1}+\frac{x}{x+1}}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x+1}{x+1} marotabaga ko'paytirish.
1+\frac{1}{\frac{x+1+x}{x+1}}=x
\frac{x+1}{x+1} va \frac{x}{x+1} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
1+\frac{1}{\frac{2x+1}{x+1}}=x
x+1+x kabi iboralarga o‘xshab birlashtiring.
1+\frac{x+1}{2x+1}=x
x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{2x+1}{x+1} ga bo'lish 1 ga k'paytirish \frac{2x+1}{x+1} ga qaytarish.
\frac{2x+1}{2x+1}+\frac{x+1}{2x+1}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{2x+1}{2x+1} marotabaga ko'paytirish.
\frac{2x+1+x+1}{2x+1}=x
\frac{2x+1}{2x+1} va \frac{x+1}{2x+1} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
\frac{3x+2}{2x+1}=x
2x+1+x+1 kabi iboralarga o‘xshab birlashtiring.
\frac{3x+2}{2x+1}-x=0
Ikkala tarafdan x ni ayirish.
\frac{3x+2}{2x+1}-\frac{x\left(2x+1\right)}{2x+1}=0
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x ni \frac{2x+1}{2x+1} marotabaga ko'paytirish.
\frac{3x+2-x\left(2x+1\right)}{2x+1}=0
\frac{3x+2}{2x+1} va \frac{x\left(2x+1\right)}{2x+1} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.
\frac{3x+2-2x^{2}-x}{2x+1}=0
3x+2-x\left(2x+1\right) ichidagi ko‘paytirishlarni bajaring.
\frac{2x+2-2x^{2}}{2x+1}=0
3x+2-2x^{2}-x kabi iboralarga o‘xshab birlashtiring.
2x+2-2x^{2}=0
x qiymati -\frac{1}{2} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x+1 ga ko'paytirish.
-2x^{2}+2x+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=\frac{-2±\sqrt{2^{2}-4\left(-2\right)\times 2}}{2\left(-2\right)}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 2 ni b va 2 ni c bilan almashtiring.
x=\frac{-2±\sqrt{4-4\left(-2\right)\times 2}}{2\left(-2\right)}
2 kvadratini chiqarish.
x=\frac{-2±\sqrt{4+8\times 2}}{2\left(-2\right)}
-4 ni -2 marotabaga ko'paytirish.
x=\frac{-2±\sqrt{4+16}}{2\left(-2\right)}
8 ni 2 marotabaga ko'paytirish.
x=\frac{-2±\sqrt{20}}{2\left(-2\right)}
4 ni 16 ga qo'shish.
x=\frac{-2±2\sqrt{5}}{2\left(-2\right)}
20 ning kvadrat ildizini chiqarish.
x=\frac{-2±2\sqrt{5}}{-4}
2 ni -2 marotabaga ko'paytirish.
x=\frac{2\sqrt{5}-2}{-4}
x=\frac{-2±2\sqrt{5}}{-4} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{5} ga qo'shish.
x=\frac{1-\sqrt{5}}{2}
-2+2\sqrt{5} ni -4 ga bo'lish.
x=\frac{-2\sqrt{5}-2}{-4}
x=\frac{-2±2\sqrt{5}}{-4} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{5} ni ayirish.
x=\frac{\sqrt{5}+1}{2}
-2-2\sqrt{5} ni -4 ga bo'lish.
x=\frac{1-\sqrt{5}}{2} x=\frac{\sqrt{5}+1}{2}
Tenglama yechildi.
1+\frac{1}{1+\frac{1}{\frac{x}{x}+\frac{1}{x}}}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x}{x} marotabaga ko'paytirish.
1+\frac{1}{1+\frac{1}{\frac{x+1}{x}}}=x
\frac{x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
1+\frac{1}{1+\frac{x}{x+1}}=x
x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{x+1}{x} ga bo'lish 1 ga k'paytirish \frac{x+1}{x} ga qaytarish.
1+\frac{1}{\frac{x+1}{x+1}+\frac{x}{x+1}}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x+1}{x+1} marotabaga ko'paytirish.
1+\frac{1}{\frac{x+1+x}{x+1}}=x
\frac{x+1}{x+1} va \frac{x}{x+1} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
1+\frac{1}{\frac{2x+1}{x+1}}=x
x+1+x kabi iboralarga o‘xshab birlashtiring.
1+\frac{x+1}{2x+1}=x
x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{2x+1}{x+1} ga bo'lish 1 ga k'paytirish \frac{2x+1}{x+1} ga qaytarish.
\frac{2x+1}{2x+1}+\frac{x+1}{2x+1}=x
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{2x+1}{2x+1} marotabaga ko'paytirish.
\frac{2x+1+x+1}{2x+1}=x
\frac{2x+1}{2x+1} va \frac{x+1}{2x+1} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.
\frac{3x+2}{2x+1}=x
2x+1+x+1 kabi iboralarga o‘xshab birlashtiring.
\frac{3x+2}{2x+1}-x=0
Ikkala tarafdan x ni ayirish.
\frac{3x+2}{2x+1}-\frac{x\left(2x+1\right)}{2x+1}=0
Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x ni \frac{2x+1}{2x+1} marotabaga ko'paytirish.
\frac{3x+2-x\left(2x+1\right)}{2x+1}=0
\frac{3x+2}{2x+1} va \frac{x\left(2x+1\right)}{2x+1} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.
\frac{3x+2-2x^{2}-x}{2x+1}=0
3x+2-x\left(2x+1\right) ichidagi ko‘paytirishlarni bajaring.
\frac{2x+2-2x^{2}}{2x+1}=0
3x+2-2x^{2}-x kabi iboralarga o‘xshab birlashtiring.
2x+2-2x^{2}=0
x qiymati -\frac{1}{2} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x+1 ga ko'paytirish.
2x-2x^{2}=-2
Ikkala tarafdan 2 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.
-2x^{2}+2x=-2
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{-2x^{2}+2x}{-2}=-\frac{2}{-2}
Ikki tarafini -2 ga bo‘ling.
x^{2}+\frac{2}{-2}x=-\frac{2}{-2}
-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.
x^{2}-x=-\frac{2}{-2}
2 ni -2 ga bo'lish.
x^{2}-x=1
-2 ni -2 ga bo'lish.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1+\left(-\frac{1}{2}\right)^{2}
-1 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{2} olish uchun. Keyin, -\frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}-x+\frac{1}{4}=1+\frac{1}{4}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.
x^{2}-x+\frac{1}{4}=\frac{5}{4}
1 ni \frac{1}{4} ga qo'shish.
\left(x-\frac{1}{2}\right)^{2}=\frac{5}{4}
x^{2}-x+\frac{1}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-\frac{1}{2}=\frac{\sqrt{5}}{2} x-\frac{1}{2}=-\frac{\sqrt{5}}{2}
Qisqartirish.
x=\frac{\sqrt{5}+1}{2} x=\frac{1-\sqrt{5}}{2}
\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.