(x-30)(162-3x)=420 eching

$$(x-30)(162-3x)=420$$

x uchun yechish

$$x=40$$

$$x=44$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-30 ga 162-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$252x-3x^{2}-4860=420$$

Ikkala tarafdan 420 ni ayirish.

$$252x-3x^{2}-4860-420=0$$

-5280 olish uchun -4860 dan 420 ni ayirish.

$$252x-3x^{2}-5280=0$$

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.

$$-3x^{2}+252x-5280=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 252 ni b va -5280 ni c bilan almashtiring.

$$x=\frac{-252±\sqrt{252^{2}-4\left(-3\right)\left(-5280\right)}}{2\left(-3\right)}$$

252 kvadratini chiqarish.

$$x=\frac{-252±\sqrt{63504-4\left(-3\right)\left(-5280\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-252±\sqrt{63504+12\left(-5280\right)}}{2\left(-3\right)}$$

12 ni -5280 marotabaga ko'paytirish.

$$x=\frac{-252±\sqrt{63504-63360}}{2\left(-3\right)}$$

63504 ni -63360 ga qo'shish.

$$x=\frac{-252±\sqrt{144}}{2\left(-3\right)}$$

144 ning kvadrat ildizini chiqarish.

$$x=\frac{-252±12}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-252±12}{-6}$$

x=\frac{-252±12}{-6} tenglamasini yeching, bunda ± musbat. -252 ni 12 ga qo'shish.

$$x=-\frac{240}{-6}$$

-240 ni -6 ga bo'lish.

$$x=40$$

x=\frac{-252±12}{-6} tenglamasini yeching, bunda ± manfiy. -252 dan 12 ni ayirish.

$$x=-\frac{264}{-6}$$

-264 ni -6 ga bo'lish.

$$x=44$$

Tenglama yechildi.

$$x=40 x=44$$

Kvadratni yakunlash bo'yicha qadamlar

x-30 ga 162-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$252x-3x^{2}-4860=420$$

4860 ni ikki tarafga qo’shing.

$$252x-3x^{2}=420+4860$$

5280 olish uchun 420 va 4860'ni qo'shing.

$$252x-3x^{2}=5280$$

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

$$-3x^{2}+252x=5280$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+252x}{-3}=\frac{5280}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{252}{-3}x=\frac{5280}{-3}$$

252 ni -3 ga bo'lish.

$$x^{2}-84x=\frac{5280}{-3}$$

5280 ni -3 ga bo'lish.

$$x^{2}-84x=-1760$$

-84 ni bo‘lish, x shartining koeffitsienti, 2 ga -42 olish uchun. Keyin, -42 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-84x+\left(-42\right)^{2}=-1760+\left(-42\right)^{2}$$

-42 kvadratini chiqarish.

$$x^{2}-84x+1764=-1760+1764$$

-1760 ni 1764 ga qo'shish.

$$x^{2}-84x+1764=4$$

x^{2}-84x+1764 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-42\right)^{2}=4$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-42\right)^{2}}=\sqrt{4}$$

Qisqartirish.

$$x-42=2 x-42=-2$$

42 ni tenglamaning ikkala tarafiga qo'shish.

$$x=44 x=40$$