left(3x-5right)^2-left(4x-2right)^2=4(9-10x) eching

$${ \left(3x-5 \right) }^{ 2 } - { \left(4x-2 \right) }^{ 2 } =4(9-10x)$$

x uchun yechish

$$x=3$$

$$x=\frac{5}{7}\approx 0,714285714$$

Guruhlash orqali faktorlash qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-5\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(4x-2\right)^{2}=4\left(9-10x\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-2\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(16x^{2}-16x+4\right)=4\left(9-10x\right)$$

16x^{2}-16x+4 teskarisini topish uchun har birining teskarisini toping.

$$9x^{2}-30x+25-16x^{2}+16x-4=4\left(9-10x\right)$$

-7x^{2} ni olish uchun 9x^{2} va -16x^{2} ni birlashtirish.

$$-7x^{2}-30x+25+16x-4=4\left(9-10x\right)$$

-14x ni olish uchun -30x va 16x ni birlashtirish.

$$-7x^{2}-14x+25-4=4\left(9-10x\right)$$

21 olish uchun 25 dan 4 ni ayirish.

$$-7x^{2}-14x+21=4\left(9-10x\right)$$

4 ga 9-10x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-7x^{2}-14x+21=36-40x$$

Ikkala tarafdan 36 ni ayirish.

$$-7x^{2}-14x+21-36=-40x$$

-15 olish uchun 21 dan 36 ni ayirish.

$$-7x^{2}-14x-15=-40x$$

40x ni ikki tarafga qo’shing.

$$-7x^{2}-14x-15+40x=0$$

26x ni olish uchun -14x va 40x ni birlashtirish.

$$-7x^{2}+26x-15=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -7x^{2}+ax+bx-15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=26 ab=-7\left(-15\right)=105$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 105-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,105 3,35 5,21 7,15$$

Har bir juftlik yigʻindisini hisoblang.

$$1+105=106 3+35=38 5+21=26 7+15=22$$

Yechim – 26 yigʻindisini beruvchi juftlik.

$$a=21 b=5$$

-7x^{2}+26x-15 ni \left(-7x^{2}+21x\right)+\left(5x-15\right) sifatida qaytadan yozish.

$$\left(-7x^{2}+21x\right)+\left(5x-15\right)$$

Birinchi guruhda 7x ni va ikkinchi guruhda -5 ni faktordan chiqaring.

$$7x\left(-x+3\right)-5\left(-x+3\right)$$

Distributiv funktsiyasidan foydalangan holda -x+3 umumiy terminini chiqaring.

$$\left(-x+3\right)\left(7x-5\right)$$

Tenglamani yechish uchun -x+3=0 va 7x-5=0 ni yeching.

$$x=3 x=\frac{5}{7}$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-5\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(4x-2\right)^{2}=4\left(9-10x\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-2\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(16x^{2}-16x+4\right)=4\left(9-10x\right)$$

16x^{2}-16x+4 teskarisini topish uchun har birining teskarisini toping.

$$9x^{2}-30x+25-16x^{2}+16x-4=4\left(9-10x\right)$$

-7x^{2} ni olish uchun 9x^{2} va -16x^{2} ni birlashtirish.

$$-7x^{2}-30x+25+16x-4=4\left(9-10x\right)$$

-14x ni olish uchun -30x va 16x ni birlashtirish.

$$-7x^{2}-14x+25-4=4\left(9-10x\right)$$

21 olish uchun 25 dan 4 ni ayirish.

$$-7x^{2}-14x+21=4\left(9-10x\right)$$

4 ga 9-10x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-7x^{2}-14x+21=36-40x$$

Ikkala tarafdan 36 ni ayirish.

$$-7x^{2}-14x+21-36=-40x$$

-15 olish uchun 21 dan 36 ni ayirish.

$$-7x^{2}-14x-15=-40x$$

40x ni ikki tarafga qo’shing.

$$-7x^{2}-14x-15+40x=0$$

26x ni olish uchun -14x va 40x ni birlashtirish.

$$-7x^{2}+26x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -7 ni a, 26 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-26±\sqrt{26^{2}-4\left(-7\right)\left(-15\right)}}{2\left(-7\right)}$$

26 kvadratini chiqarish.

$$x=\frac{-26±\sqrt{676-4\left(-7\right)\left(-15\right)}}{2\left(-7\right)}$$

-4 ni -7 marotabaga ko'paytirish.

$$x=\frac{-26±\sqrt{676+28\left(-15\right)}}{2\left(-7\right)}$$

28 ni -15 marotabaga ko'paytirish.

$$x=\frac{-26±\sqrt{676-420}}{2\left(-7\right)}$$

676 ni -420 ga qo'shish.

$$x=\frac{-26±\sqrt{256}}{2\left(-7\right)}$$

256 ning kvadrat ildizini chiqarish.

$$x=\frac{-26±16}{2\left(-7\right)}$$

2 ni -7 marotabaga ko'paytirish.

$$x=\frac{-26±16}{-14}$$

x=\frac{-26±16}{-14} tenglamasini yeching, bunda ± musbat. -26 ni 16 ga qo'shish.

$$x=-\frac{10}{-14}$$

\frac{-10}{-14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{7}$$

x=\frac{-26±16}{-14} tenglamasini yeching, bunda ± manfiy. -26 dan 16 ni ayirish.

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

-42 ni -14 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=\frac{5}{7} x=3$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3x-5\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(4x-2\right)^{2}=4\left(9-10x\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-2\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}-30x+25-\left(16x^{2}-16x+4\right)=4\left(9-10x\right)$$

16x^{2}-16x+4 teskarisini topish uchun har birining teskarisini toping.

$$9x^{2}-30x+25-16x^{2}+16x-4=4\left(9-10x\right)$$

-7x^{2} ni olish uchun 9x^{2} va -16x^{2} ni birlashtirish.

$$-7x^{2}-30x+25+16x-4=4\left(9-10x\right)$$

-14x ni olish uchun -30x va 16x ni birlashtirish.

$$-7x^{2}-14x+25-4=4\left(9-10x\right)$$

21 olish uchun 25 dan 4 ni ayirish.

$$-7x^{2}-14x+21=4\left(9-10x\right)$$

4 ga 9-10x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-7x^{2}-14x+21=36-40x$$

40x ni ikki tarafga qo’shing.

$$-7x^{2}-14x+21+40x=36$$

26x ni olish uchun -14x va 40x ni birlashtirish.

$$-7x^{2}+26x+21=36$$

Ikkala tarafdan 21 ni ayirish.

$$-7x^{2}+26x=36-21$$

15 olish uchun 36 dan 21 ni ayirish.

$$-7x^{2}+26x=15$$

Ikki tarafini -7 ga bo‘ling.

$$\frac{-7x^{2}+26x}{-7}=\frac{15}{-7}$$

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

$$x^{2}+\frac{26}{-7}x=\frac{15}{-7}$$

26 ni -7 ga bo'lish.

$$x^{2}-\frac{26}{7}x=\frac{15}{-7}$$

15 ni -7 ga bo'lish.

$$x^{2}-\frac{26}{7}x=-\frac{15}{7}$$

-\frac{26}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{7} olish uchun. Keyin, -\frac{13}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{26}{7}x+\left(-\frac{13}{7}\right)^{2}=-\frac{15}{7}+\left(-\frac{13}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{7} kvadratini chiqarish.

$$x^{2}-\frac{26}{7}x+\frac{169}{49}=-\frac{15}{7}+\frac{169}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{15}{7} ni \frac{169}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{26}{7}x+\frac{169}{49}=\frac{64}{49}$$

x^{2}-\frac{26}{7}x+\frac{169}{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.

$$\left(x-\frac{13}{7}\right)^{2}=\frac{64}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{7}\right)^{2}}=\sqrt{\frac{64}{49}}$$

Qisqartirish.

$$x-\frac{13}{7}=\frac{8}{7} x-\frac{13}{7}=-\frac{8}{7}$$

\frac{13}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=\frac{5}{7}$$