18x^2+33x=180 eching

$$18 { x }^{ 2 } +33x=180$$

x uchun yechish

$$x = \frac{\sqrt{1561} - 11}{12} \approx 2,375791044$$

$$x=\frac{-\sqrt{1561}-11}{12}\approx -4,209124378$$

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.

$$18x^{2}+33x=180$$

Tenglamaning ikkala tarafidan 180 ni ayirish.

$$18x^{2}+33x-180=180-180$$

O‘zidan 180 ayirilsa 0 qoladi.

$$18x^{2}+33x-180=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 18 ni a, 33 ni b va -180 ni c bilan almashtiring.

$$x=\frac{-33±\sqrt{33^{2}-4\times 18\left(-180\right)}}{2\times 18}$$

33 kvadratini chiqarish.

$$x=\frac{-33±\sqrt{1089-4\times 18\left(-180\right)}}{2\times 18}$$

-4 ni 18 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089-72\left(-180\right)}}{2\times 18}$$

-72 ni -180 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089+12960}}{2\times 18}$$

1089 ni 12960 ga qo'shish.

$$x=\frac{-33±\sqrt{14049}}{2\times 18}$$

14049 ning kvadrat ildizini chiqarish.

$$x=\frac{-33±3\sqrt{1561}}{2\times 18}$$

2 ni 18 marotabaga ko'paytirish.

$$x=\frac{-33±3\sqrt{1561}}{36}$$

x=\frac{-33±3\sqrt{1561}}{36} tenglamasini yeching, bunda ± musbat. -33 ni 3\sqrt{1561} ga qo'shish.

$$x=\frac{3\sqrt{1561}-33}{36}$$

-33+3\sqrt{1561} ni 36 ga bo'lish.

$$x=\frac{\sqrt{1561}-11}{12}$$

x=\frac{-33±3\sqrt{1561}}{36} tenglamasini yeching, bunda ± manfiy. -33 dan 3\sqrt{1561} ni ayirish.

$$x=\frac{-3\sqrt{1561}-33}{36}$$

-33-3\sqrt{1561} ni 36 ga bo'lish.

$$x=\frac{-\sqrt{1561}-11}{12}$$

Tenglama yechildi.

$$x=\frac{\sqrt{1561}-11}{12} x=\frac{-\sqrt{1561}-11}{12}$$

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.

$$18x^{2}+33x=180$$

Ikki tarafini 18 ga bo‘ling.

$$\frac{18x^{2}+33x}{18}=\frac{180}{18}$$

18 ga bo'lish 18 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{33}{18}x=\frac{180}{18}$$

\frac{33}{18} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{11}{6}x=\frac{180}{18}$$

180 ni 18 ga bo'lish.

$$x^{2}+\frac{11}{6}x=10$$

\frac{11}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{12} olish uchun. Keyin, \frac{11}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=10+\left(\frac{11}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{12} kvadratini chiqarish.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=10+\frac{121}{144}$$

10 ni \frac{121}{144} ga qo'shish.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{1561}{144}$$

x^{2}+\frac{11}{6}x+\frac{121}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{1561}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{1561}{144}}$$

Qisqartirish.

$$x+\frac{11}{12}=\frac{\sqrt{1561}}{12} x+\frac{11}{12}=-\frac{\sqrt{1561}}{12}$$

Tenglamaning ikkala tarafidan \frac{11}{12} ni ayirish.

$$x=\frac{\sqrt{1561}-11}{12} x=\frac{-\sqrt{1561}-11}{12}$$