left(m-4right)^2-4mleft(m+1right)=0 eching

$${ \left(m-4 \right) }^{ 2 } -4m \left( m+1 \right) =0$$

m uchun yechish

$$m=\frac{2\sqrt{21}}{3}-2\approx 1,055050463$$

$$m=-\frac{2\sqrt{21}}{3}-2\approx -5,055050463$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(m-4\right)^{2} kengaytirilishi uchun ishlating.

$$m^{2}-8m+16-4m\left(m+1\right)=0$$

-4m ga m+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$m^{2}-8m+16-4m^{2}-4m=0$$

-3m^{2} ni olish uchun m^{2} va -4m^{2} ni birlashtirish.

$$-3m^{2}-8m+16-4m=0$$

-12m ni olish uchun -8m va -4m ni birlashtirish.

$$-3m^{2}-12m+16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -12 ni b va 16 ni c bilan almashtiring.

$$m=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

-12 kvadratini chiqarish.

$$m=\frac{-\left(-12\right)±\sqrt{144-4\left(-3\right)\times 16}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$m=\frac{-\left(-12\right)±\sqrt{144+12\times 16}}{2\left(-3\right)}$$

12 ni 16 marotabaga ko'paytirish.

$$m=\frac{-\left(-12\right)±\sqrt{144+192}}{2\left(-3\right)}$$

144 ni 192 ga qo'shish.

$$m=\frac{-\left(-12\right)±\sqrt{336}}{2\left(-3\right)}$$

336 ning kvadrat ildizini chiqarish.

$$m=\frac{-\left(-12\right)±4\sqrt{21}}{2\left(-3\right)}$$

-12 ning teskarisi 12 ga teng.

$$m=\frac{12±4\sqrt{21}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$m=\frac{12±4\sqrt{21}}{-6}$$

m=\frac{12±4\sqrt{21}}{-6} tenglamasini yeching, bunda ± musbat. 12 ni 4\sqrt{21} ga qo'shish.

$$m=\frac{4\sqrt{21}+12}{-6}$$

12+4\sqrt{21} ni -6 ga bo'lish.

$$m=-\frac{2\sqrt{21}}{3}-2$$

m=\frac{12±4\sqrt{21}}{-6} tenglamasini yeching, bunda ± manfiy. 12 dan 4\sqrt{21} ni ayirish.

$$m=\frac{12-4\sqrt{21}}{-6}$$

12-4\sqrt{21} ni -6 ga bo'lish.

$$m=\frac{2\sqrt{21}}{3}-2$$

Tenglama yechildi.

$$m=-\frac{2\sqrt{21}}{3}-2 m=\frac{2\sqrt{21}}{3}-2$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(m-4\right)^{2} kengaytirilishi uchun ishlating.

$$m^{2}-8m+16-4m\left(m+1\right)=0$$

-4m ga m+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$m^{2}-8m+16-4m^{2}-4m=0$$

-3m^{2} ni olish uchun m^{2} va -4m^{2} ni birlashtirish.

$$-3m^{2}-8m+16-4m=0$$

-12m ni olish uchun -8m va -4m ni birlashtirish.

$$-3m^{2}-12m+16=0$$

Ikkala tarafdan 16 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-3m^{2}-12m=-16$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3m^{2}-12m}{-3}=-\frac{16}{-3}$$

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

$$m^{2}+\left(-\frac{12}{-3}\right)m=-\frac{16}{-3}$$

-12 ni -3 ga bo'lish.

$$m^{2}+4m=-\frac{16}{-3}$$

-16 ni -3 ga bo'lish.

$$m^{2}+4m=\frac{16}{3}$$

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

$$m^{2}+4m+2^{2}=\frac{16}{3}+2^{2}$$

2 kvadratini chiqarish.

$$m^{2}+4m+4=\frac{16}{3}+4$$

\frac{16}{3} ni 4 ga qo'shish.

$$m^{2}+4m+4=\frac{28}{3}$$

m^{2}+4m+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.

$$\left(m+2\right)^{2}=\frac{28}{3}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(m+2\right)^{2}}=\sqrt{\frac{28}{3}}$$

Qisqartirish.

$$m+2=\frac{2\sqrt{21}}{3} m+2=-\frac{2\sqrt{21}}{3}$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$m=\frac{2\sqrt{21}}{3}-2 m=-\frac{2\sqrt{21}}{3}-2$$