8x^2+72x+108=0 eching

$$8 x ^ { 2 } + 72 x + 108 = 0$$

x uchun yechish

$$x=\frac{3\sqrt{3}-9}{2}\approx -1,901923789$$

$$x=\frac{-3\sqrt{3}-9}{2}\approx -7,098076211$$

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.

$$8x^{2}+72x+108=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 72 ni b va 108 ni c bilan almashtiring.

$$x=\frac{-72±\sqrt{72^{2}-4\times 8\times 108}}{2\times 8}$$

72 kvadratini chiqarish.

$$x=\frac{-72±\sqrt{5184-4\times 8\times 108}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-72±\sqrt{5184-32\times 108}}{2\times 8}$$

-32 ni 108 marotabaga ko'paytirish.

$$x=\frac{-72±\sqrt{5184-3456}}{2\times 8}$$

5184 ni -3456 ga qo'shish.

$$x=\frac{-72±\sqrt{1728}}{2\times 8}$$

1728 ning kvadrat ildizini chiqarish.

$$x=\frac{-72±24\sqrt{3}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{-72±24\sqrt{3}}{16}$$

x=\frac{-72±24\sqrt{3}}{16} tenglamasini yeching, bunda ± musbat. -72 ni 24\sqrt{3} ga qo'shish.

$$x=\frac{24\sqrt{3}-72}{16}$$

-72+24\sqrt{3} ni 16 ga bo'lish.

$$x=\frac{3\sqrt{3}-9}{2}$$

x=\frac{-72±24\sqrt{3}}{16} tenglamasini yeching, bunda ± manfiy. -72 dan 24\sqrt{3} ni ayirish.

$$x=\frac{-24\sqrt{3}-72}{16}$$

-72-24\sqrt{3} ni 16 ga bo'lish.

$$x=\frac{-3\sqrt{3}-9}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{3}-9}{2} x=\frac{-3\sqrt{3}-9}{2}$$

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.

$$8x^{2}+72x+108=0$$

Tenglamaning ikkala tarafidan 108 ni ayirish.

$$8x^{2}+72x+108-108=-108$$

O‘zidan 108 ayirilsa 0 qoladi.

$$8x^{2}+72x=-108$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}+72x}{8}=-\frac{108}{8}$$

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

$$x^{2}+\frac{72}{8}x=-\frac{108}{8}$$

72 ni 8 ga bo'lish.

$$x^{2}+9x=-\frac{108}{8}$$

\frac{-108}{8} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+9x=-\frac{27}{2}$$

9 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{2} olish uchun. Keyin, \frac{9}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=-\frac{27}{2}+\left(\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{2} kvadratini chiqarish.

$$x^{2}+9x+\frac{81}{4}=-\frac{27}{2}+\frac{81}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{27}{2} ni \frac{81}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+9x+\frac{81}{4}=\frac{27}{4}$$

x^{2}+9x+\frac{81}{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.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{27}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{27}{4}}$$

Qisqartirish.

$$x+\frac{9}{2}=\frac{3\sqrt{3}}{2} x+\frac{9}{2}=-\frac{3\sqrt{3}}{2}$$

Tenglamaning ikkala tarafidan \frac{9}{2} ni ayirish.

$$x=\frac{3\sqrt{3}-9}{2} x=\frac{-3\sqrt{3}-9}{2}$$