8x^2-x-180=0 eching

$$8 x ^ { 2 } - x - 180 = 0$$

x uchun yechish

$$x = \frac{\sqrt{5761} + 1}{16} \approx 4,806328227$$

$$x=\frac{1-\sqrt{5761}}{16}\approx -4,681328227$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$8x^{2}-x-180=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -1 ni b va -180 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 8\left(-180\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-32\left(-180\right)}}{2\times 8}$$

-32 ni -180 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+5760}}{2\times 8}$$

1 ni 5760 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{5761}}{2\times 8}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{5761}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{5761}}{16}$$

x=\frac{1±\sqrt{5761}}{16} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{5761} ga qo'shish.

$$x=\frac{\sqrt{5761}+1}{16}$$

x=\frac{1±\sqrt{5761}}{16} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{5761} ni ayirish.

$$x=\frac{1-\sqrt{5761}}{16}$$

Tenglama yechildi.

$$x=\frac{\sqrt{5761}+1}{16} x=\frac{1-\sqrt{5761}}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$8x^{2}-x-180=0$$

180 ni tenglamaning ikkala tarafiga qo'shish.

$$8x^{2}-x-180-\left(-180\right)=-\left(-180\right)$$

O‘zidan -180 ayirilsa 0 qoladi.

$$8x^{2}-x=-\left(-180\right)$$

0 dan -180 ni ayirish.

$$8x^{2}-x=180$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}-x}{8}=\frac{180}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{1}{8}x=\frac{180}{8}$$

\frac{180}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{8}x=\frac{45}{2}$$

-\frac{1}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{16} olish uchun. Keyin, -\frac{1}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{8}x+\left(-\frac{1}{16}\right)^{2}=\frac{45}{2}+\left(-\frac{1}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{16} kvadratini chiqarish.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{45}{2}+\frac{1}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{45}{2} ni \frac{1}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{8}x+\frac{1}{256}=\frac{5761}{256}$$

x^{2}-\frac{1}{8}x+\frac{1}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{16}\right)^{2}=\frac{5761}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{16}\right)^{2}}=\sqrt{\frac{5761}{256}}$$

Qisqartirish.

$$x-\frac{1}{16}=\frac{\sqrt{5761}}{16} x-\frac{1}{16}=-\frac{\sqrt{5761}}{16}$$

\frac{1}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{5761}+1}{16} x=\frac{1-\sqrt{5761}}{16}$$