10x^2+10x+8=3x^2-10x+11 eching

$$10 { x }^{ 2 } +10x+8=3 { x }^{ 2 } -10x+11$$

x uchun yechish

$$x=-3$$

$$x=\frac{1}{7}\approx 0,142857143$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 3x^{2} ni ayirish.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

7x^{2} ni olish uchun 10x^{2} va -3x^{2} ni birlashtirish.

$$7x^{2}+10x+8=-10x+11$$

10x ni ikki tarafga qo’shing.

$$7x^{2}+10x+8+10x=11$$

20x ni olish uchun 10x va 10x ni birlashtirish.

$$7x^{2}+20x+8=11$$

Ikkala tarafdan 11 ni ayirish.

$$7x^{2}+20x+8-11=0$$

-3 olish uchun 8 dan 11 ni ayirish.

$$7x^{2}+20x-3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 7x^{2}+ax+bx-3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=20 ab=7\left(-3\right)=-21$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -21-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,21 -3,7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+21=20 -3+7=4$$

Yechim – 20 yigʻindisini beruvchi juftlik.

$$a=-1 b=21$$

7x^{2}+20x-3 ni \left(7x^{2}-x\right)+\left(21x-3\right) sifatida qaytadan yozish.

$$\left(7x^{2}-x\right)+\left(21x-3\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$x\left(7x-1\right)+3\left(7x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 7x-1 umumiy terminini chiqaring.

$$\left(7x-1\right)\left(x+3\right)$$

Tenglamani yechish uchun 7x-1=0 va x+3=0 ni yeching.

$$x=\frac{1}{7} x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3x^{2} ni ayirish.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

7x^{2} ni olish uchun 10x^{2} va -3x^{2} ni birlashtirish.

$$7x^{2}+10x+8=-10x+11$$

10x ni ikki tarafga qo’shing.

$$7x^{2}+10x+8+10x=11$$

20x ni olish uchun 10x va 10x ni birlashtirish.

$$7x^{2}+20x+8=11$$

Ikkala tarafdan 11 ni ayirish.

$$7x^{2}+20x+8-11=0$$

-3 olish uchun 8 dan 11 ni ayirish.

$$7x^{2}+20x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, 20 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\times 7\left(-3\right)}}{2\times 7}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\times 7\left(-3\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400-28\left(-3\right)}}{2\times 7}$$

-28 ni -3 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+84}}{2\times 7}$$

400 ni 84 ga qo'shish.

$$x=\frac{-20±\sqrt{484}}{2\times 7}$$

484 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±22}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-20±22}{14}$$

x=\frac{-20±22}{14} tenglamasini yeching, bunda ± musbat. -20 ni 22 ga qo'shish.

$$x=\frac{2}{14}$$

\frac{2}{14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{7}$$

x=\frac{-20±22}{14} tenglamasini yeching, bunda ± manfiy. -20 dan 22 ni ayirish.

$$x=-\frac{42}{14}$$

-42 ni 14 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=\frac{1}{7} x=-3$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3x^{2} ni ayirish.

$$10x^{2}+10x+8-3x^{2}=-10x+11$$

7x^{2} ni olish uchun 10x^{2} va -3x^{2} ni birlashtirish.

$$7x^{2}+10x+8=-10x+11$$

10x ni ikki tarafga qo’shing.

$$7x^{2}+10x+8+10x=11$$

20x ni olish uchun 10x va 10x ni birlashtirish.

$$7x^{2}+20x+8=11$$

Ikkala tarafdan 8 ni ayirish.

$$7x^{2}+20x=11-8$$

3 olish uchun 11 dan 8 ni ayirish.

$$7x^{2}+20x=3$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}+20x}{7}=\frac{3}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{20}{7}x=\frac{3}{7}$$

\frac{20}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{10}{7} olish uchun. Keyin, \frac{10}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{20}{7}x+\left(\frac{10}{7}\right)^{2}=\frac{3}{7}+\left(\frac{10}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{10}{7} kvadratini chiqarish.

$$x^{2}+\frac{20}{7}x+\frac{100}{49}=\frac{3}{7}+\frac{100}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{7} ni \frac{100}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{20}{7}x+\frac{100}{49}=\frac{121}{49}$$

x^{2}+\frac{20}{7}x+\frac{100}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{10}{7}\right)^{2}=\frac{121}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{10}{7}\right)^{2}}=\sqrt{\frac{121}{49}}$$

Qisqartirish.

$$x+\frac{10}{7}=\frac{11}{7} x+\frac{10}{7}=-\frac{11}{7}$$

Tenglamaning ikkala tarafidan \frac{10}{7} ni ayirish.

$$x=\frac{1}{7} x=-3$$