x^2-5x+6/x^2-7x+12=2 eching

$$\frac{ { x }^{ 2 } -5x+6 }{ { x }^{ 2 } -7x+12 } =2$$

x uchun yechish

$$x=6$$

Guruhlash orqali faktorlash qadamlari

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga ko'paytirish.

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

2 ga x-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

2x-8 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-5x+6=2x^{2}-14x+24$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

14x ni ikki tarafga qo’shing.

$$-x^{2}-5x+6+14x=24$$

9x ni olish uchun -5x va 14x ni birlashtirish.

$$-x^{2}+9x+6=24$$

Ikkala tarafdan 24 ni ayirish.

$$-x^{2}+9x+6-24=0$$

-18 olish uchun 6 dan 24 ni ayirish.

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

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx-18 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=9 ab=-\left(-18\right)=18$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 18-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,18 2,9 3,6$$

Har bir juftlik yigʻindisini hisoblang.

$$1+18=19 2+9=11 3+6=9$$

Yechim – 9 yigʻindisini beruvchi juftlik.

$$a=6 b=3$$

-x^{2}+9x-18 ni \left(-x^{2}+6x\right)+\left(3x-18\right) sifatida qaytadan yozish.

$$\left(-x^{2}+6x\right)+\left(3x-18\right)$$

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

$$-x\left(x-6\right)+3\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(-x+3\right)$$

Tenglamani yechish uchun x-6=0 va -x+3=0 ni yeching.

$$x=6 x=3$$

x qiymati 3 teng bo‘lmaydi.

$$x=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga ko'paytirish.

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

2 ga x-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

2x-8 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-5x+6=2x^{2}-14x+24$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

14x ni ikki tarafga qo’shing.

$$-x^{2}-5x+6+14x=24$$

9x ni olish uchun -5x va 14x ni birlashtirish.

$$-x^{2}+9x+6=24$$

Ikkala tarafdan 24 ni ayirish.

$$-x^{2}+9x+6-24=0$$

-18 olish uchun 6 dan 24 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 9 ni b va -18 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+4\left(-18\right)}}{2\left(-1\right)}$$

4 ni -18 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-72}}{2\left(-1\right)}$$

81 ni -72 ga qo'shish.

$$x=\frac{-9±\sqrt{9}}{2\left(-1\right)}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-9±3}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-9±3}{-2}$$

x=\frac{-9±3}{-2} tenglamasini yeching, bunda ± musbat. -9 ni 3 ga qo'shish.

$$x=-\frac{6}{-2}$$

-6 ni -2 ga bo'lish.

$$x=3$$

x=\frac{-9±3}{-2} tenglamasini yeching, bunda ± manfiy. -9 dan 3 ni ayirish.

$$x=-\frac{12}{-2}$$

-12 ni -2 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=3 x=6$$

x qiymati 3 teng bo‘lmaydi.

$$x=6$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 3,4 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-4\right)\left(x-3\right) ga ko'paytirish.

$$x^{2}-5x+6=2\left(x-4\right)\left(x-3\right)$$

2 ga x-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-5x+6=\left(2x-8\right)\left(x-3\right)$$

2x-8 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-5x+6=2x^{2}-14x+24$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-5x+6-2x^{2}=-14x+24$$

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

$$-x^{2}-5x+6=-14x+24$$

14x ni ikki tarafga qo’shing.

$$-x^{2}-5x+6+14x=24$$

9x ni olish uchun -5x va 14x ni birlashtirish.

$$-x^{2}+9x+6=24$$

Ikkala tarafdan 6 ni ayirish.

$$-x^{2}+9x=24-6$$

18 olish uchun 24 dan 6 ni ayirish.

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

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+9x}{-1}=\frac{18}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{9}{-1}x=\frac{18}{-1}$$

9 ni -1 ga bo'lish.

$$x^{2}-9x=\frac{18}{-1}$$

18 ni -1 ga bo'lish.

$$x^{2}-9x=-18$$

-9 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{2} olish uchun. Keyin, -\frac{9}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-18+\left(-\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{2} kvadratini chiqarish.

$$x^{2}-9x+\frac{81}{4}=-18+\frac{81}{4}$$

-18 ni \frac{81}{4} ga qo'shish.

$$x^{2}-9x+\frac{81}{4}=\frac{9}{4}$$

x^{2}-9x+\frac{81}{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.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-\frac{9}{2}=\frac{3}{2} x-\frac{9}{2}=-\frac{3}{2}$$

\frac{9}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=3$$

x qiymati 3 teng bo‘lmaydi.

$$x=6$$