(30-3x)(10-2x)=216 eching

$$(30-3x)(10-2x)=216$$

x uchun yechish

$$x=1$$

$$x=14$$

Kvadrat tenglamalaridan foydalanish qoidalari

30-3x ga 10-2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$300-90x+6x^{2}=216$$

Ikkala tarafdan 216 ni ayirish.

$$300-90x+6x^{2}-216=0$$

84 olish uchun 300 dan 216 ni ayirish.

$$84-90x+6x^{2}=0$$

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.

$$6x^{2}-90x+84=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -90 ni b va 84 ni c bilan almashtiring.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 6\times 84}}{2\times 6}$$

-90 kvadratini chiqarish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 6\times 84}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-24\times 84}}{2\times 6}$$

-24 ni 84 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-2016}}{2\times 6}$$

8100 ni -2016 ga qo'shish.

$$x=\frac{-\left(-90\right)±\sqrt{6084}}{2\times 6}$$

6084 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-90\right)±78}{2\times 6}$$

-90 ning teskarisi 90 ga teng.

$$x=\frac{90±78}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{90±78}{12}$$

x=\frac{90±78}{12} tenglamasini yeching, bunda ± musbat. 90 ni 78 ga qo'shish.

$$x=\frac{168}{12}$$

168 ni 12 ga bo'lish.

$$x=14$$

x=\frac{90±78}{12} tenglamasini yeching, bunda ± manfiy. 90 dan 78 ni ayirish.

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

12 ni 12 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=14 x=1$$

Kvadratni yakunlash bo'yicha qadamlar

30-3x ga 10-2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$300-90x+6x^{2}=216$$

Ikkala tarafdan 300 ni ayirish.

$$-90x+6x^{2}=216-300$$

-84 olish uchun 216 dan 300 ni ayirish.

$$-90x+6x^{2}=-84$$

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

$$6x^{2}-90x=-84$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-90x}{6}=-\frac{84}{6}$$

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

$$x^{2}+\left(-\frac{90}{6}\right)x=-\frac{84}{6}$$

-90 ni 6 ga bo'lish.

$$x^{2}-15x=-\frac{84}{6}$$

-84 ni 6 ga bo'lish.

$$x^{2}-15x=-14$$

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

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-14+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-14+\frac{225}{4}$$

-14 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{169}{4}$$

x^{2}-15x+\frac{225}{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.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{169}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{13}{2} x-\frac{15}{2}=-\frac{13}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=14 x=1$$