(x-2)^2-9(x+1)^2=0 eching

$$( x - 2 ) ^ { 2 } - 9 ( x + 1 ) ^ { 2 } = 0$$

x uchun yechish

$$x=-\frac{1}{4}=-0,25$$

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

Guruhlash orqali faktorlash qadamlari

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

$$x^{2}-4x+4-9\left(x+1\right)^{2}=0$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-9\left(x^{2}+2x+1\right)=0$$

-9 ga x^{2}+2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x+4-9x^{2}-18x-9=0$$

-8x^{2} ni olish uchun x^{2} va -9x^{2} ni birlashtirish.

$$-8x^{2}-4x+4-18x-9=0$$

-22x ni olish uchun -4x va -18x ni birlashtirish.

$$-8x^{2}-22x+4-9=0$$

-5 olish uchun 4 dan 9 ni ayirish.

$$-8x^{2}-22x-5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -8x^{2}+ax+bx-5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-22 ab=-8\left(-5\right)=40$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 40-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Yechim – -22 yigʻindisini beruvchi juftlik.

$$a=-2 b=-20$$

-8x^{2}-22x-5 ni \left(-8x^{2}-2x\right)+\left(-20x-5\right) sifatida qaytadan yozish.

$$\left(-8x^{2}-2x\right)+\left(-20x-5\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$2x\left(-4x-1\right)+5\left(-4x-1\right)$$

Distributiv funktsiyasidan foydalangan holda -4x-1 umumiy terminini chiqaring.

$$\left(-4x-1\right)\left(2x+5\right)$$

Tenglamani yechish uchun -4x-1=0 va 2x+5=0 ni yeching.

$$x=-\frac{1}{4} x=-\frac{5}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}-4x+4-9\left(x+1\right)^{2}=0$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-9\left(x^{2}+2x+1\right)=0$$

-9 ga x^{2}+2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x+4-9x^{2}-18x-9=0$$

-8x^{2} ni olish uchun x^{2} va -9x^{2} ni birlashtirish.

$$-8x^{2}-4x+4-18x-9=0$$

-22x ni olish uchun -4x va -18x ni birlashtirish.

$$-8x^{2}-22x+4-9=0$$

-5 olish uchun 4 dan 9 ni ayirish.

$$-8x^{2}-22x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, -22 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\left(-8\right)\left(-5\right)}}{2\left(-8\right)}$$

-22 kvadratini chiqarish.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\left(-8\right)\left(-5\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-\left(-22\right)±\sqrt{484+32\left(-5\right)}}{2\left(-8\right)}$$

32 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-22\right)±\sqrt{484-160}}{2\left(-8\right)}$$

484 ni -160 ga qo'shish.

$$x=\frac{-\left(-22\right)±\sqrt{324}}{2\left(-8\right)}$$

324 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-22\right)±18}{2\left(-8\right)}$$

-22 ning teskarisi 22 ga teng.

$$x=\frac{22±18}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{22±18}{-16}$$

x=\frac{22±18}{-16} tenglamasini yeching, bunda ± musbat. 22 ni 18 ga qo'shish.

$$x=\frac{40}{-16}$$

\frac{40}{-16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{5}{2}$$

x=\frac{22±18}{-16} tenglamasini yeching, bunda ± manfiy. 22 dan 18 ni ayirish.

$$x=\frac{4}{-16}$$

\frac{4}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=-\frac{5}{2} x=-\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x^{2}-4x+4-9\left(x+1\right)^{2}=0$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-4x+4-9\left(x^{2}+2x+1\right)=0$$

-9 ga x^{2}+2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-4x+4-9x^{2}-18x-9=0$$

-8x^{2} ni olish uchun x^{2} va -9x^{2} ni birlashtirish.

$$-8x^{2}-4x+4-18x-9=0$$

-22x ni olish uchun -4x va -18x ni birlashtirish.

$$-8x^{2}-22x+4-9=0$$

-5 olish uchun 4 dan 9 ni ayirish.

$$-8x^{2}-22x-5=0$$

5 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-8x^{2}-22x=5$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}-22x}{-8}=\frac{5}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{22}{-8}\right)x=\frac{5}{-8}$$

\frac{-22}{-8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{11}{4}x=\frac{5}{-8}$$

5 ni -8 ga bo'lish.

$$x^{2}+\frac{11}{4}x=-\frac{5}{8}$$

\frac{11}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{8} olish uchun. Keyin, \frac{11}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{11}{4}x+\left(\frac{11}{8}\right)^{2}=-\frac{5}{8}+\left(\frac{11}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{8} kvadratini chiqarish.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=-\frac{5}{8}+\frac{121}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{8} ni \frac{121}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{11}{4}x+\frac{121}{64}=\frac{81}{64}$$

x^{2}+\frac{11}{4}x+\frac{121}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{11}{8}\right)^{2}=\frac{81}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Qisqartirish.

$$x+\frac{11}{8}=\frac{9}{8} x+\frac{11}{8}=-\frac{9}{8}$$

Tenglamaning ikkala tarafidan \frac{11}{8} ni ayirish.

$$x=-\frac{1}{4} x=-\frac{5}{2}$$