4x+102=-20x(3-6x) eching

$$4 x + 102 = - 20 x ( 3 - 6 x )$$

x uchun yechish

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15}\approx 1,226412003$$

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15}\approx -0,69307867$$

Kvadrat tenglamalaridan foydalanish qoidalari

-20x ga 3-6x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+102=-60x+120x^{2}$$

60x ni ikki tarafga qo’shing.

$$4x+102+60x=120x^{2}$$

64x ni olish uchun 4x va 60x ni birlashtirish.

$$64x+102=120x^{2}$$

Ikkala tarafdan 120x^{2} ni ayirish.

$$64x+102-120x^{2}=0$$

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.

$$-120x^{2}+64x+102=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -120 ni a, 64 ni b va 102 ni c bilan almashtiring.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-120\right)\times 102}}{2\left(-120\right)}$$

64 kvadratini chiqarish.

$$x=\frac{-64±\sqrt{4096-4\left(-120\right)\times 102}}{2\left(-120\right)}$$

-4 ni -120 marotabaga ko'paytirish.

$$x=\frac{-64±\sqrt{4096+480\times 102}}{2\left(-120\right)}$$

480 ni 102 marotabaga ko'paytirish.

$$x=\frac{-64±\sqrt{4096+48960}}{2\left(-120\right)}$$

4096 ni 48960 ga qo'shish.

$$x=\frac{-64±\sqrt{53056}}{2\left(-120\right)}$$

53056 ning kvadrat ildizini chiqarish.

$$x=\frac{-64±8\sqrt{829}}{2\left(-120\right)}$$

2 ni -120 marotabaga ko'paytirish.

$$x=\frac{-64±8\sqrt{829}}{-240}$$

x=\frac{-64±8\sqrt{829}}{-240} tenglamasini yeching, bunda ± musbat. -64 ni 8\sqrt{829} ga qo'shish.

$$x=\frac{8\sqrt{829}-64}{-240}$$

-64+8\sqrt{829} ni -240 ga bo'lish.

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15}$$

x=\frac{-64±8\sqrt{829}}{-240} tenglamasini yeching, bunda ± manfiy. -64 dan 8\sqrt{829} ni ayirish.

$$x=\frac{-8\sqrt{829}-64}{-240}$$

-64-8\sqrt{829} ni -240 ga bo'lish.

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15}$$

Tenglama yechildi.

$$x=-\frac{\sqrt{829}}{30}+\frac{4}{15} x=\frac{\sqrt{829}}{30}+\frac{4}{15}$$

Kvadratni yakunlash bo'yicha qadamlar

-20x ga 3-6x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x+102=-60x+120x^{2}$$

60x ni ikki tarafga qo’shing.

$$4x+102+60x=120x^{2}$$

64x ni olish uchun 4x va 60x ni birlashtirish.

$$64x+102=120x^{2}$$

Ikkala tarafdan 120x^{2} ni ayirish.

$$64x+102-120x^{2}=0$$

Ikkala tarafdan 102 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$64x-120x^{2}=-102$$

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

$$-120x^{2}+64x=-102$$

Ikki tarafini -120 ga bo‘ling.

$$\frac{-120x^{2}+64x}{-120}=-\frac{102}{-120}$$

-120 ga bo'lish -120 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{64}{-120}x=-\frac{102}{-120}$$

\frac{64}{-120} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{15}x=-\frac{102}{-120}$$

\frac{-102}{-120} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{15}x=\frac{17}{20}$$

-\frac{8}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{15} olish uchun. Keyin, -\frac{4}{15} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=\frac{17}{20}+\left(-\frac{4}{15}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{15} kvadratini chiqarish.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{17}{20}+\frac{16}{225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{17}{20} ni \frac{16}{225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{829}{900}$$

x^{2}-\frac{8}{15}x+\frac{16}{225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{15}\right)^{2}=\frac{829}{900}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{\frac{829}{900}}$$

Qisqartirish.

$$x-\frac{4}{15}=\frac{\sqrt{829}}{30} x-\frac{4}{15}=-\frac{\sqrt{829}}{30}$$

\frac{4}{15} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{829}}{30}+\frac{4}{15} x=-\frac{\sqrt{829}}{30}+\frac{4}{15}$$