2(x-7)^2-37=-35 eching

$$2 ( x - 7 ) ^ { 2 } - 37 = - 35$$

x uchun yechish

$$x=8$$

$$x=6$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-7\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}-14x+49\right)-37=-35$$

2 ga x^{2}-14x+49 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-28x+98-37=-35$$

61 olish uchun 98 dan 37 ni ayirish.

$$2x^{2}-28x+61=-35$$

35 ni ikki tarafga qo’shing.

$$2x^{2}-28x+61+35=0$$

96 olish uchun 61 va 35'ni qo'shing.

$$2x^{2}-28x+96=0$$

Ikki tarafini 2 ga bo‘ling.

$$x^{2}-14x+48=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+48 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-14 ab=1\times 48=48$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 48-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

Yechim – -14 yigʻindisini beruvchi juftlik.

$$a=-8 b=-6$$

x^{2}-14x+48 ni \left(x^{2}-8x\right)+\left(-6x+48\right) sifatida qaytadan yozish.

$$\left(x^{2}-8x\right)+\left(-6x+48\right)$$

Birinchi guruhda x ni va ikkinchi guruhda -6 ni faktordan chiqaring.

$$x\left(x-8\right)-6\left(x-8\right)$$

Distributiv funktsiyasidan foydalangan holda x-8 umumiy terminini chiqaring.

$$\left(x-8\right)\left(x-6\right)$$

Tenglamani yechish uchun x-8=0 va x-6=0 ni yeching.

$$x=8 x=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-7\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}-14x+49\right)-37=-35$$

2 ga x^{2}-14x+49 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-28x+98-37=-35$$

61 olish uchun 98 dan 37 ni ayirish.

$$2x^{2}-28x+61=-35$$

35 ni ikki tarafga qo’shing.

$$2x^{2}-28x+61+35=0$$

96 olish uchun 61 va 35'ni qo'shing.

$$2x^{2}-28x+96=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -28 ni b va 96 ni c bilan almashtiring.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 96}}{2\times 2}$$

-28 kvadratini chiqarish.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 96}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 96}}{2\times 2}$$

-8 ni 96 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784-768}}{2\times 2}$$

784 ni -768 ga qo'shish.

$$x=\frac{-\left(-28\right)±\sqrt{16}}{2\times 2}$$

16 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-28\right)±4}{2\times 2}$$

-28 ning teskarisi 28 ga teng.

$$x=\frac{28±4}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{28±4}{4}$$

x=\frac{28±4}{4} tenglamasini yeching, bunda ± musbat. 28 ni 4 ga qo'shish.

$$x=\frac{32}{4}$$

32 ni 4 ga bo'lish.

$$x=8$$

x=\frac{28±4}{4} tenglamasini yeching, bunda ± manfiy. 28 dan 4 ni ayirish.

$$x=\frac{24}{4}$$

24 ni 4 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=8 x=6$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-7\right)^{2} kengaytirilishi uchun ishlating.

$$2\left(x^{2}-14x+49\right)-37=-35$$

2 ga x^{2}-14x+49 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-28x+98-37=-35$$

61 olish uchun 98 dan 37 ni ayirish.

$$2x^{2}-28x+61=-35$$

Ikkala tarafdan 61 ni ayirish.

$$2x^{2}-28x=-35-61$$

-96 olish uchun -35 dan 61 ni ayirish.

$$2x^{2}-28x=-96$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-28x}{2}=-\frac{96}{2}$$

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

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{96}{2}$$

-28 ni 2 ga bo'lish.

$$x^{2}-14x=-\frac{96}{2}$$

-96 ni 2 ga bo'lish.

$$x^{2}-14x=-48$$

-14 ni bo‘lish, x shartining koeffitsienti, 2 ga -7 olish uchun. Keyin, -7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-14x+\left(-7\right)^{2}=-48+\left(-7\right)^{2}$$

-7 kvadratini chiqarish.

$$x^{2}-14x+49=-48+49$$

-48 ni 49 ga qo'shish.

$$x^{2}-14x+49=1$$

x^{2}-14x+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.

$$\left(x-7\right)^{2}=1$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{1}$$

Qisqartirish.

$$x-7=1 x-7=-1$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=6$$