2x-1/x=3 eching

$$2x- \frac{ 1 }{ x } = 3$$

x uchun yechish

$$x = \frac{\sqrt{17} + 3}{4} \approx 1,780776406$$

$$x=\frac{3-\sqrt{17}}{4}\approx -0,280776406$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x ga ko'paytirish.

$$2xx-1=3x$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$2x^{2}-1=3x$$

Ikkala tarafdan 3x ni ayirish.

$$2x^{2}-1-3x=0$$

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.

$$2x^{2}-3x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-1\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-1\right)}}{2\times 2}$$

-8 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+8}}{2\times 2}$$

9 ni 8 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{17}}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{17}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{17}}{4}$$

x=\frac{3±\sqrt{17}}{4} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}+3}{4}$$

x=\frac{3±\sqrt{17}}{4} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{17} ni ayirish.

$$x=\frac{3-\sqrt{17}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{17}+3}{4} x=\frac{3-\sqrt{17}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x ga ko'paytirish.

$$2xx-1=3x$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$2x^{2}-1=3x$$

Ikkala tarafdan 3x ni ayirish.

$$2x^{2}-1-3x=0$$

1 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$2x^{2}-3x=1$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-3x}{2}=\frac{1}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{3}{2}x=\frac{1}{2}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{1}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{17}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{17}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{\sqrt{17}}{4} x-\frac{3}{4}=-\frac{\sqrt{17}}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{17}+3}{4} x=\frac{3-\sqrt{17}}{4}$$