4x^2-25=(2x-5)(7x+3) eching

$$4 x ^ { 2 } - 25 = ( 2 x - 5 ) ( 7 x + 3 )$$

x uchun yechish

$$x=\frac{2}{5}=0,4$$

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

Guruhlash orqali faktorlash qadamlari

2x-5 ga 7x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-25=14x^{2}-29x-15$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$4x^{2}-25-14x^{2}=-29x-15$$

-10x^{2} ni olish uchun 4x^{2} va -14x^{2} ni birlashtirish.

$$-10x^{2}-25=-29x-15$$

29x ni ikki tarafga qo’shing.

$$-10x^{2}-25+29x=-15$$

15 ni ikki tarafga qo’shing.

$$-10x^{2}-25+29x+15=0$$

-10 olish uchun -25 va 15'ni qo'shing.

$$-10x^{2}-10+29x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-10x^{2}+29x-10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -10x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=29 ab=-10\left(-10\right)=100$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 100-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,100 2,50 4,25 5,20 10,10$$

Har bir juftlik yigʻindisini hisoblang.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

Yechim – 29 yigʻindisini beruvchi juftlik.

$$a=25 b=4$$

-10x^{2}+29x-10 ni \left(-10x^{2}+25x\right)+\left(4x-10\right) sifatida qaytadan yozish.

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

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

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

Distributiv funktsiyasidan foydalangan holda 2x-5 umumiy terminini chiqaring.

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

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

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

Kvadrat tenglamalaridan foydalanish qoidalari

2x-5 ga 7x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-25=14x^{2}-29x-15$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$4x^{2}-25-14x^{2}=-29x-15$$

-10x^{2} ni olish uchun 4x^{2} va -14x^{2} ni birlashtirish.

$$-10x^{2}-25=-29x-15$$

29x ni ikki tarafga qo’shing.

$$-10x^{2}-25+29x=-15$$

15 ni ikki tarafga qo’shing.

$$-10x^{2}-25+29x+15=0$$

-10 olish uchun -25 va 15'ni qo'shing.

$$-10x^{2}-10+29x=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-10x^{2}+29x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -10 ni a, 29 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-29±\sqrt{29^{2}-4\left(-10\right)\left(-10\right)}}{2\left(-10\right)}$$

29 kvadratini chiqarish.

$$x=\frac{-29±\sqrt{841-4\left(-10\right)\left(-10\right)}}{2\left(-10\right)}$$

-4 ni -10 marotabaga ko'paytirish.

$$x=\frac{-29±\sqrt{841+40\left(-10\right)}}{2\left(-10\right)}$$

40 ni -10 marotabaga ko'paytirish.

$$x=\frac{-29±\sqrt{841-400}}{2\left(-10\right)}$$

841 ni -400 ga qo'shish.

$$x=\frac{-29±\sqrt{441}}{2\left(-10\right)}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-29±21}{2\left(-10\right)}$$

2 ni -10 marotabaga ko'paytirish.

$$x=\frac{-29±21}{-20}$$

x=\frac{-29±21}{-20} tenglamasini yeching, bunda ± musbat. -29 ni 21 ga qo'shish.

$$x=-\frac{8}{-20}$$

\frac{-8}{-20} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-29±21}{-20} tenglamasini yeching, bunda ± manfiy. -29 dan 21 ni ayirish.

$$x=-\frac{50}{-20}$$

\frac{-50}{-20} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

2x-5 ga 7x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$4x^{2}-25=14x^{2}-29x-15$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$4x^{2}-25-14x^{2}=-29x-15$$

-10x^{2} ni olish uchun 4x^{2} va -14x^{2} ni birlashtirish.

$$-10x^{2}-25=-29x-15$$

29x ni ikki tarafga qo’shing.

$$-10x^{2}-25+29x=-15$$

25 ni ikki tarafga qo’shing.

$$-10x^{2}+29x=-15+25$$

10 olish uchun -15 va 25'ni qo'shing.

$$-10x^{2}+29x=10$$

Ikki tarafini -10 ga bo‘ling.

$$\frac{-10x^{2}+29x}{-10}=\frac{10}{-10}$$

-10 ga bo'lish -10 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{29}{-10}x=\frac{10}{-10}$$

29 ni -10 ga bo'lish.

$$x^{2}-\frac{29}{10}x=\frac{10}{-10}$$

10 ni -10 ga bo'lish.

$$x^{2}-\frac{29}{10}x=-1$$

-\frac{29}{10} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{29}{20} olish uchun. Keyin, -\frac{29}{20} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{29}{10}x+\left(-\frac{29}{20}\right)^{2}=-1+\left(-\frac{29}{20}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{29}{20} kvadratini chiqarish.

$$x^{2}-\frac{29}{10}x+\frac{841}{400}=-1+\frac{841}{400}$$

-1 ni \frac{841}{400} ga qo'shish.

$$x^{2}-\frac{29}{10}x+\frac{841}{400}=\frac{441}{400}$$

x^{2}-\frac{29}{10}x+\frac{841}{400} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{29}{20}\right)^{2}=\frac{441}{400}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{29}{20}\right)^{2}}=\sqrt{\frac{441}{400}}$$

Qisqartirish.

$$x-\frac{29}{20}=\frac{21}{20} x-\frac{29}{20}=-\frac{21}{20}$$

\frac{29}{20} ni tenglamaning ikkala tarafiga qo'shish.

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