(3+r)^2+(15+r)^2=18^2 eching

$$( 3 + r ) ^ { 2 } + ( 15 + r ) ^ { 2 } = 18 ^ { 2 }$$

r uchun yechish

$$r=3\sqrt{14}-9\approx 2,22497216$$

$$r=-3\sqrt{14}-9\approx -20,22497216$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3+r\right)^{2} kengaytirilishi uchun ishlating.

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(15+r\right)^{2} kengaytirilishi uchun ishlating.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

234 olish uchun 9 va 225'ni qo'shing.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

36r ni olish uchun 6r va 30r ni birlashtirish.

$$234+36r+r^{2}+r^{2}=18^{2}$$

2r^{2} ni olish uchun r^{2} va r^{2} ni birlashtirish.

$$234+36r+2r^{2}=18^{2}$$

2 daraja ko‘rsatkichini 18 ga hisoblang va 324 ni qiymatni oling.

$$234+36r+2r^{2}=324$$

Ikkala tarafdan 324 ni ayirish.

$$234+36r+2r^{2}-324=0$$

-90 olish uchun 234 dan 324 ni ayirish.

$$-90+36r+2r^{2}=0$$

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.

$$2r^{2}+36r-90=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 36 ni b va -90 ni c bilan almashtiring.

$$r=\frac{-36±\sqrt{36^{2}-4\times 2\left(-90\right)}}{2\times 2}$$

36 kvadratini chiqarish.

$$r=\frac{-36±\sqrt{1296-4\times 2\left(-90\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$r=\frac{-36±\sqrt{1296-8\left(-90\right)}}{2\times 2}$$

-8 ni -90 marotabaga ko'paytirish.

$$r=\frac{-36±\sqrt{1296+720}}{2\times 2}$$

1296 ni 720 ga qo'shish.

$$r=\frac{-36±\sqrt{2016}}{2\times 2}$$

2016 ning kvadrat ildizini chiqarish.

$$r=\frac{-36±12\sqrt{14}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$r=\frac{-36±12\sqrt{14}}{4}$$

r=\frac{-36±12\sqrt{14}}{4} tenglamasini yeching, bunda ± musbat. -36 ni 12\sqrt{14} ga qo'shish.

$$r=\frac{12\sqrt{14}-36}{4}$$

-36+12\sqrt{14} ni 4 ga bo'lish.

$$r=3\sqrt{14}-9$$

r=\frac{-36±12\sqrt{14}}{4} tenglamasini yeching, bunda ± manfiy. -36 dan 12\sqrt{14} ni ayirish.

$$r=\frac{-12\sqrt{14}-36}{4}$$

-36-12\sqrt{14} ni 4 ga bo'lish.

$$r=-3\sqrt{14}-9$$

Tenglama yechildi.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3+r\right)^{2} kengaytirilishi uchun ishlating.

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(15+r\right)^{2} kengaytirilishi uchun ishlating.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

234 olish uchun 9 va 225'ni qo'shing.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

36r ni olish uchun 6r va 30r ni birlashtirish.

$$234+36r+r^{2}+r^{2}=18^{2}$$

2r^{2} ni olish uchun r^{2} va r^{2} ni birlashtirish.

$$234+36r+2r^{2}=18^{2}$$

2 daraja ko‘rsatkichini 18 ga hisoblang va 324 ni qiymatni oling.

$$234+36r+2r^{2}=324$$

Ikkala tarafdan 234 ni ayirish.

$$36r+2r^{2}=324-234$$

90 olish uchun 324 dan 234 ni ayirish.

$$36r+2r^{2}=90$$

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

$$2r^{2}+36r=90$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2r^{2}+36r}{2}=\frac{90}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$r^{2}+\frac{36}{2}r=\frac{90}{2}$$

36 ni 2 ga bo'lish.

$$r^{2}+18r=\frac{90}{2}$$

90 ni 2 ga bo'lish.

$$r^{2}+18r=45$$

18 ni bo‘lish, x shartining koeffitsienti, 2 ga 9 olish uchun. Keyin, 9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$r^{2}+18r+9^{2}=45+9^{2}$$

9 kvadratini chiqarish.

$$r^{2}+18r+81=45+81$$

45 ni 81 ga qo'shish.

$$r^{2}+18r+81=126$$

r^{2}+18r+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(r+9\right)^{2}=126$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(r+9\right)^{2}}=\sqrt{126}$$

Qisqartirish.

$$r+9=3\sqrt{14} r+9=-3\sqrt{14}$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$