(90-30x)(20x)=50 eching

$$(90-30x)(20x) = 50$$

x uchun yechish

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 2,971960144$$

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2}\approx 0,028039856$$

Kvadrat tenglamalaridan foydalanish qoidalari

90-30x ga 20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(1800-600x\right)x=50$$

1800-600x ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$1800x-600x^{2}=50$$

Ikkala tarafdan 50 ni ayirish.

$$1800x-600x^{2}-50=0$$

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.

$$-600x^{2}+1800x-50=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -600 ni a, 1800 ni b va -50 ni c bilan almashtiring.

$$x=\frac{-1800±\sqrt{1800^{2}-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

1800 kvadratini chiqarish.

$$x=\frac{-1800±\sqrt{3240000-4\left(-600\right)\left(-50\right)}}{2\left(-600\right)}$$

-4 ni -600 marotabaga ko'paytirish.

$$x=\frac{-1800±\sqrt{3240000+2400\left(-50\right)}}{2\left(-600\right)}$$

2400 ni -50 marotabaga ko'paytirish.

$$x=\frac{-1800±\sqrt{3240000-120000}}{2\left(-600\right)}$$

3240000 ni -120000 ga qo'shish.

$$x=\frac{-1800±\sqrt{3120000}}{2\left(-600\right)}$$

3120000 ning kvadrat ildizini chiqarish.

$$x=\frac{-1800±200\sqrt{78}}{2\left(-600\right)}$$

2 ni -600 marotabaga ko'paytirish.

$$x=\frac{-1800±200\sqrt{78}}{-1200}$$

x=\frac{-1800±200\sqrt{78}}{-1200} tenglamasini yeching, bunda ± musbat. -1800 ni 200\sqrt{78} ga qo'shish.

$$x=\frac{200\sqrt{78}-1800}{-1200}$$

-1800+200\sqrt{78} ni -1200 ga bo'lish.

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$

x=\frac{-1800±200\sqrt{78}}{-1200} tenglamasini yeching, bunda ± manfiy. -1800 dan 200\sqrt{78} ni ayirish.

$$x=\frac{-200\sqrt{78}-1800}{-1200}$$

-1800-200\sqrt{78} ni -1200 ga bo'lish.

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Tenglama yechildi.

$$x=-\frac{\sqrt{78}}{6}+\frac{3}{2} x=\frac{\sqrt{78}}{6}+\frac{3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

90-30x ga 20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(1800-600x\right)x=50$$

1800-600x ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$1800x-600x^{2}=50$$

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

$$-600x^{2}+1800x=50$$

Ikki tarafini -600 ga bo‘ling.

$$\frac{-600x^{2}+1800x}{-600}=\frac{50}{-600}$$

-600 ga bo'lish -600 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{1800}{-600}x=\frac{50}{-600}$$

1800 ni -600 ga bo'lish.

$$x^{2}-3x=\frac{50}{-600}$$

\frac{50}{-600} ulushini 50 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-3x=-\frac{1}{12}$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-\frac{1}{12}+\left(-\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=-\frac{1}{12}+\frac{9}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{12} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-3x+\frac{9}{4}=\frac{13}{6}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{13}{6}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{13}{6}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{78}}{6} x-\frac{3}{2}=-\frac{\sqrt{78}}{6}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{78}}{6}+\frac{3}{2} x=-\frac{\sqrt{78}}{6}+\frac{3}{2}$$