1=11+17x^2-32x eching

$$1 = 11 + 17 x ^ { 2 } - 32 x$$

x uchun yechish

$$x = \frac{\sqrt{86} + 16}{17} \approx 1,486683441$$

$$x=\frac{16-\sqrt{86}}{17}\approx 0,3956695$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$11+17x^{2}-32x=1$$

Ikkala tarafdan 1 ni ayirish.

$$11+17x^{2}-32x-1=0$$

10 olish uchun 11 dan 1 ni ayirish.

$$10+17x^{2}-32x=0$$

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.

$$17x^{2}-32x+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 17 ni a, -32 ni b va 10 ni c bilan almashtiring.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 17\times 10}}{2\times 17}$$

-32 kvadratini chiqarish.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 17\times 10}}{2\times 17}$$

-4 ni 17 marotabaga ko'paytirish.

$$x=\frac{-\left(-32\right)±\sqrt{1024-68\times 10}}{2\times 17}$$

-68 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-32\right)±\sqrt{1024-680}}{2\times 17}$$

1024 ni -680 ga qo'shish.

$$x=\frac{-\left(-32\right)±\sqrt{344}}{2\times 17}$$

344 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-32\right)±2\sqrt{86}}{2\times 17}$$

-32 ning teskarisi 32 ga teng.

$$x=\frac{32±2\sqrt{86}}{2\times 17}$$

2 ni 17 marotabaga ko'paytirish.

$$x=\frac{32±2\sqrt{86}}{34}$$

x=\frac{32±2\sqrt{86}}{34} tenglamasini yeching, bunda ± musbat. 32 ni 2\sqrt{86} ga qo'shish.

$$x=\frac{2\sqrt{86}+32}{34}$$

32+2\sqrt{86} ni 34 ga bo'lish.

$$x=\frac{\sqrt{86}+16}{17}$$

x=\frac{32±2\sqrt{86}}{34} tenglamasini yeching, bunda ± manfiy. 32 dan 2\sqrt{86} ni ayirish.

$$x=\frac{32-2\sqrt{86}}{34}$$

32-2\sqrt{86} ni 34 ga bo'lish.

$$x=\frac{16-\sqrt{86}}{17}$$

Tenglama yechildi.

$$x=\frac{\sqrt{86}+16}{17} x=\frac{16-\sqrt{86}}{17}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$11+17x^{2}-32x=1$$

Ikkala tarafdan 11 ni ayirish.

$$17x^{2}-32x=1-11$$

-10 olish uchun 1 dan 11 ni ayirish.

$$17x^{2}-32x=-10$$

Ikki tarafini 17 ga bo‘ling.

$$\frac{17x^{2}-32x}{17}=-\frac{10}{17}$$

17 ga bo'lish 17 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{32}{17}x=-\frac{10}{17}$$

-\frac{32}{17} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{16}{17} olish uchun. Keyin, -\frac{16}{17} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{32}{17}x+\left(-\frac{16}{17}\right)^{2}=-\frac{10}{17}+\left(-\frac{16}{17}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{16}{17} kvadratini chiqarish.

$$x^{2}-\frac{32}{17}x+\frac{256}{289}=-\frac{10}{17}+\frac{256}{289}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{10}{17} ni \frac{256}{289} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{32}{17}x+\frac{256}{289}=\frac{86}{289}$$

x^{2}-\frac{32}{17}x+\frac{256}{289} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{16}{17}\right)^{2}=\frac{86}{289}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{16}{17}\right)^{2}}=\sqrt{\frac{86}{289}}$$

Qisqartirish.

$$x-\frac{16}{17}=\frac{\sqrt{86}}{17} x-\frac{16}{17}=-\frac{\sqrt{86}}{17}$$

\frac{16}{17} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{86}+16}{17} x=\frac{16-\sqrt{86}}{17}$$