(k+2)^3-k^3=20 eching

$$( k + 2 ) ^ { 3 } - k ^ { 3 } = 20$$

k uchun yechish

$$k=\sqrt{3}-1\approx 0,732050808$$

$$k=-\left(\sqrt{3}+1\right)\approx -2,732050808$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$k^{3}+6k^{2}+12k+8-k^{3}=20$$

0 ni olish uchun k^{3} va -k^{3} ni birlashtirish.

$$6k^{2}+12k+8=20$$

Ikkala tarafdan 20 ni ayirish.

$$6k^{2}+12k+8-20=0$$

-12 olish uchun 8 dan 20 ni ayirish.

$$6k^{2}+12k-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, 12 ni b va -12 ni c bilan almashtiring.

$$k=\frac{-12±\sqrt{12^{2}-4\times 6\left(-12\right)}}{2\times 6}$$

12 kvadratini chiqarish.

$$k=\frac{-12±\sqrt{144-4\times 6\left(-12\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$k=\frac{-12±\sqrt{144-24\left(-12\right)}}{2\times 6}$$

-24 ni -12 marotabaga ko'paytirish.

$$k=\frac{-12±\sqrt{144+288}}{2\times 6}$$

144 ni 288 ga qo'shish.

$$k=\frac{-12±\sqrt{432}}{2\times 6}$$

432 ning kvadrat ildizini chiqarish.

$$k=\frac{-12±12\sqrt{3}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$k=\frac{-12±12\sqrt{3}}{12}$$

k=\frac{-12±12\sqrt{3}}{12} tenglamasini yeching, bunda ± musbat. -12 ni 12\sqrt{3} ga qo'shish.

$$k=\frac{12\sqrt{3}-12}{12}$$

-12+12\sqrt{3} ni 12 ga bo'lish.

$$k=\sqrt{3}-1$$

k=\frac{-12±12\sqrt{3}}{12} tenglamasini yeching, bunda ± manfiy. -12 dan 12\sqrt{3} ni ayirish.

$$k=\frac{-12\sqrt{3}-12}{12}$$

-12-12\sqrt{3} ni 12 ga bo'lish.

$$k=-\sqrt{3}-1$$

Tenglama yechildi.

$$k=\sqrt{3}-1 k=-\sqrt{3}-1$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$k^{3}+6k^{2}+12k+8-k^{3}=20$$

0 ni olish uchun k^{3} va -k^{3} ni birlashtirish.

$$6k^{2}+12k+8=20$$

Ikkala tarafdan 8 ni ayirish.

$$6k^{2}+12k=20-8$$

12 olish uchun 20 dan 8 ni ayirish.

$$6k^{2}+12k=12$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6k^{2}+12k}{6}=\frac{12}{6}$$

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

$$k^{2}+\frac{12}{6}k=\frac{12}{6}$$

12 ni 6 ga bo'lish.

$$k^{2}+2k=\frac{12}{6}$$

12 ni 6 ga bo'lish.

$$k^{2}+2k=2$$

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}+2k+1^{2}=2+1^{2}$$

1 kvadratini chiqarish.

$$k^{2}+2k+1=2+1$$

2 ni 1 ga qo'shish.

$$k^{2}+2k+1=3$$

k^{2}+2k+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k+1\right)^{2}=3$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k+1\right)^{2}}=\sqrt{3}$$

Qisqartirish.

$$k+1=\sqrt{3} k+1=-\sqrt{3}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$k=\sqrt{3}-1 k=-\sqrt{3}-1$$