5x^2-20x+12=x^2+7x-6 eching

$$5 x ^ { 2 } - 20 x + 12 = x ^ { 2 } + 7 x - 6$$

x uchun yechish

$$x=\frac{3}{4}=0,75$$

$$x=6$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan x^{2} ni ayirish.

$$5x^{2}-20x+12-x^{2}=7x-6$$

4x^{2} ni olish uchun 5x^{2} va -x^{2} ni birlashtirish.

$$4x^{2}-20x+12=7x-6$$

Ikkala tarafdan 7x ni ayirish.

$$4x^{2}-20x+12-7x=-6$$

-27x ni olish uchun -20x va -7x ni birlashtirish.

$$4x^{2}-27x+12=-6$$

6 ni ikki tarafga qo’shing.

$$4x^{2}-27x+12+6=0$$

18 olish uchun 12 va 6'ni qo'shing.

$$4x^{2}-27x+18=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 4x^{2}+ax+bx+18 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-27 ab=4\times 18=72$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 72-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

Yechim – -27 yigʻindisini beruvchi juftlik.

$$a=-24 b=-3$$

4x^{2}-27x+18 ni \left(4x^{2}-24x\right)+\left(-3x+18\right) sifatida qaytadan yozish.

$$\left(4x^{2}-24x\right)+\left(-3x+18\right)$$

Birinchi guruhda 4x ni va ikkinchi guruhda -3 ni faktordan chiqaring.

$$4x\left(x-6\right)-3\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(4x-3\right)$$

Tenglamani yechish uchun x-6=0 va 4x-3=0 ni yeching.

$$x=6 x=\frac{3}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$5x^{2}-20x+12-x^{2}=7x-6$$

4x^{2} ni olish uchun 5x^{2} va -x^{2} ni birlashtirish.

$$4x^{2}-20x+12=7x-6$$

Ikkala tarafdan 7x ni ayirish.

$$4x^{2}-20x+12-7x=-6$$

-27x ni olish uchun -20x va -7x ni birlashtirish.

$$4x^{2}-27x+12=-6$$

6 ni ikki tarafga qo’shing.

$$4x^{2}-27x+12+6=0$$

18 olish uchun 12 va 6'ni qo'shing.

$$4x^{2}-27x+18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -27 ni b va 18 ni c bilan almashtiring.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 4\times 18}}{2\times 4}$$

-27 kvadratini chiqarish.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\times 4\times 18}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-27\right)±\sqrt{729-16\times 18}}{2\times 4}$$

-16 ni 18 marotabaga ko'paytirish.

$$x=\frac{-\left(-27\right)±\sqrt{729-288}}{2\times 4}$$

729 ni -288 ga qo'shish.

$$x=\frac{-\left(-27\right)±\sqrt{441}}{2\times 4}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-27\right)±21}{2\times 4}$$

-27 ning teskarisi 27 ga teng.

$$x=\frac{27±21}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{27±21}{8}$$

x=\frac{27±21}{8} tenglamasini yeching, bunda ± musbat. 27 ni 21 ga qo'shish.

$$x=\frac{48}{8}$$

48 ni 8 ga bo'lish.

$$x=6$$

x=\frac{27±21}{8} tenglamasini yeching, bunda ± manfiy. 27 dan 21 ni ayirish.

$$x=\frac{6}{8}$$

\frac{6}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{3}{4}$$

Tenglama yechildi.

$$x=6 x=\frac{3}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$5x^{2}-20x+12-x^{2}=7x-6$$

4x^{2} ni olish uchun 5x^{2} va -x^{2} ni birlashtirish.

$$4x^{2}-20x+12=7x-6$$

Ikkala tarafdan 7x ni ayirish.

$$4x^{2}-20x+12-7x=-6$$

-27x ni olish uchun -20x va -7x ni birlashtirish.

$$4x^{2}-27x+12=-6$$

Ikkala tarafdan 12 ni ayirish.

$$4x^{2}-27x=-6-12$$

-18 olish uchun -6 dan 12 ni ayirish.

$$4x^{2}-27x=-18$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-27x}{4}=-\frac{18}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{27}{4}x=-\frac{18}{4}$$

\frac{-18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{27}{4}x=-\frac{9}{2}$$

-\frac{27}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{27}{8} olish uchun. Keyin, -\frac{27}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{27}{4}x+\left(-\frac{27}{8}\right)^{2}=-\frac{9}{2}+\left(-\frac{27}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{27}{8} kvadratini chiqarish.

$$x^{2}-\frac{27}{4}x+\frac{729}{64}=-\frac{9}{2}+\frac{729}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{2} ni \frac{729}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{27}{4}x+\frac{729}{64}=\frac{441}{64}$$

x^{2}-\frac{27}{4}x+\frac{729}{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.

$$\left(x-\frac{27}{8}\right)^{2}=\frac{441}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{27}{8}\right)^{2}}=\sqrt{\frac{441}{64}}$$

Qisqartirish.

$$x-\frac{27}{8}=\frac{21}{8} x-\frac{27}{8}=-\frac{21}{8}$$

\frac{27}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=\frac{3}{4}$$