2x^2+11x+9=10x+10 eching

$$2 x ^ { 2 } + 11 x + 9 = 10 x + 10$$

x uchun yechish

$$x=-1$$

$$x=\frac{1}{2}=0,5$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 10x ni ayirish.

$$2x^{2}+11x+9-10x=10$$

x ni olish uchun 11x va -10x ni birlashtirish.

$$2x^{2}+x+9=10$$

Ikkala tarafdan 10 ni ayirish.

$$2x^{2}+x+9-10=0$$

-1 olish uchun 9 dan 10 ni ayirish.

$$2x^{2}+x-1=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=1 ab=2\left(-1\right)=-2$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=-1 b=2$$

2x^{2}+x-1 ni \left(2x^{2}-x\right)+\left(2x-1\right) sifatida qaytadan yozish.

$$\left(2x^{2}-x\right)+\left(2x-1\right)$$

2x^{2}-x ichida x ni ajrating.

$$x\left(2x-1\right)+2x-1$$

Distributiv funktsiyasidan foydalangan holda 2x-1 umumiy terminini chiqaring.

$$\left(2x-1\right)\left(x+1\right)$$

Tenglamani yechish uchun 2x-1=0 va x+1=0 ni yeching.

$$x=\frac{1}{2} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 10x ni ayirish.

$$2x^{2}+11x+9-10x=10$$

x ni olish uchun 11x va -10x ni birlashtirish.

$$2x^{2}+x+9=10$$

Ikkala tarafdan 10 ni ayirish.

$$2x^{2}+x+9-10=0$$

-1 olish uchun 9 dan 10 ni ayirish.

$$2x^{2}+x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 1 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-1\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-8\left(-1\right)}}{2\times 2}$$

-8 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+8}}{2\times 2}$$

1 ni 8 ga qo'shish.

$$x=\frac{-1±\sqrt{9}}{2\times 2}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-1±3}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-1±3}{4}$$

x=\frac{-1±3}{4} tenglamasini yeching, bunda ± musbat. -1 ni 3 ga qo'shish.

$$x=\frac{2}{4}$$

\frac{2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{2}$$

x=\frac{-1±3}{4} tenglamasini yeching, bunda ± manfiy. -1 dan 3 ni ayirish.

$$x=-\frac{4}{4}$$

-4 ni 4 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=\frac{1}{2} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 10x ni ayirish.

$$2x^{2}+11x+9-10x=10$$

x ni olish uchun 11x va -10x ni birlashtirish.

$$2x^{2}+x+9=10$$

Ikkala tarafdan 9 ni ayirish.

$$2x^{2}+x=10-9$$

1 olish uchun 10 dan 9 ni ayirish.

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

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+x}{2}=\frac{1}{2}$$

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

$$x^{2}+\frac{1}{2}x=\frac{1}{2}$$

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{1}{2}+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}$$

x^{2}+\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{9}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Qisqartirish.

$$x+\frac{1}{4}=\frac{3}{4} x+\frac{1}{4}=-\frac{3}{4}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$x=\frac{1}{2} x=-1$$