3x(11-2x)=15 eching

$$3x(11-2x)=15$$

x uchun yechish

$$x=5$$

$$x=\frac{1}{2}=0,5$$

Kvadrat tenglamalaridan foydalanish qoidalari

3x ga 11-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$33x-6x^{2}=15$$

Ikkala tarafdan 15 ni ayirish.

$$33x-6x^{2}-15=0$$

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.

$$-6x^{2}+33x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, 33 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-33±\sqrt{33^{2}-4\left(-6\right)\left(-15\right)}}{2\left(-6\right)}$$

33 kvadratini chiqarish.

$$x=\frac{-33±\sqrt{1089-4\left(-6\right)\left(-15\right)}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089+24\left(-15\right)}}{2\left(-6\right)}$$

24 ni -15 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089-360}}{2\left(-6\right)}$$

1089 ni -360 ga qo'shish.

$$x=\frac{-33±\sqrt{729}}{2\left(-6\right)}$$

729 ning kvadrat ildizini chiqarish.

$$x=\frac{-33±27}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{-33±27}{-12}$$

x=\frac{-33±27}{-12} tenglamasini yeching, bunda ± musbat. -33 ni 27 ga qo'shish.

$$x=-\frac{6}{-12}$$

\frac{-6}{-12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{2}$$

x=\frac{-33±27}{-12} tenglamasini yeching, bunda ± manfiy. -33 dan 27 ni ayirish.

$$x=-\frac{60}{-12}$$

-60 ni -12 ga bo'lish.

$$x=5$$

Tenglama yechildi.

$$x=\frac{1}{2} x=5$$

Kvadratni yakunlash bo'yicha qadamlar

3x ga 11-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$33x-6x^{2}=15$$

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

$$-6x^{2}+33x=15$$

Ikki tarafini -6 ga bo‘ling.

$$\frac{-6x^{2}+33x}{-6}=\frac{15}{-6}$$

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{33}{-6}x=\frac{15}{-6}$$

\frac{33}{-6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{11}{2}x=\frac{15}{-6}$$

\frac{15}{-6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{11}{2}x=-\frac{5}{2}$$

-\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{4} olish uchun. Keyin, -\frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{5}{2}+\left(-\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{4} kvadratini chiqarish.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{5}{2}+\frac{121}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{2} ni \frac{121}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{81}{16}$$

x^{2}-\frac{11}{2}x+\frac{121}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{81}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Qisqartirish.

$$x-\frac{11}{4}=\frac{9}{4} x-\frac{11}{4}=-\frac{9}{4}$$

\frac{11}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=5 x=\frac{1}{2}$$