25left(4+xright)^2+7(5-x)(5+x)=295-45x^2 eching

$$25 { \left(4+x \right) }^{ 2 } +7(5-x)(5+x)=295-45 { x }^{ 2 }$$

x uchun yechish (complex solution)

$$x=\frac{-100+2\sqrt{1910}i}{63}\approx -1,587301587+1,387414183i$$

$$x=\frac{-2\sqrt{1910}i-100}{63}\approx -1,587301587-1,387414183i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$25\left(16+8x+x^{2}\right)+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

25 ga 16+8x+x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400+200x+25x^{2}+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

7 ga 5-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400+200x+25x^{2}+\left(35-7x\right)\left(5+x\right)=295-45x^{2}$$

35-7x ga 5+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$400+200x+25x^{2}+175-7x^{2}=295-45x^{2}$$

575 olish uchun 400 va 175'ni qo'shing.

$$575+200x+25x^{2}-7x^{2}=295-45x^{2}$$

18x^{2} ni olish uchun 25x^{2} va -7x^{2} ni birlashtirish.

$$575+200x+18x^{2}=295-45x^{2}$$

Ikkala tarafdan 295 ni ayirish.

$$575+200x+18x^{2}-295=-45x^{2}$$

280 olish uchun 575 dan 295 ni ayirish.

$$280+200x+18x^{2}=-45x^{2}$$

45x^{2} ni ikki tarafga qo’shing.

$$280+200x+18x^{2}+45x^{2}=0$$

63x^{2} ni olish uchun 18x^{2} va 45x^{2} ni birlashtirish.

$$280+200x+63x^{2}=0$$

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.

$$63x^{2}+200x+280=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 63 ni a, 200 ni b va 280 ni c bilan almashtiring.

$$x=\frac{-200±\sqrt{200^{2}-4\times 63\times 280}}{2\times 63}$$

200 kvadratini chiqarish.

$$x=\frac{-200±\sqrt{40000-4\times 63\times 280}}{2\times 63}$$

-4 ni 63 marotabaga ko'paytirish.

$$x=\frac{-200±\sqrt{40000-252\times 280}}{2\times 63}$$

-252 ni 280 marotabaga ko'paytirish.

$$x=\frac{-200±\sqrt{40000-70560}}{2\times 63}$$

40000 ni -70560 ga qo'shish.

$$x=\frac{-200±\sqrt{-30560}}{2\times 63}$$

-30560 ning kvadrat ildizini chiqarish.

$$x=\frac{-200±4\sqrt{1910}i}{2\times 63}$$

2 ni 63 marotabaga ko'paytirish.

$$x=\frac{-200±4\sqrt{1910}i}{126}$$

x=\frac{-200±4\sqrt{1910}i}{126} tenglamasini yeching, bunda ± musbat. -200 ni 4i\sqrt{1910} ga qo'shish.

$$x=\frac{-200+4\sqrt{1910}i}{126}$$

-200+4i\sqrt{1910} ni 126 ga bo'lish.

$$x=\frac{-100+2\sqrt{1910}i}{63}$$

x=\frac{-200±4\sqrt{1910}i}{126} tenglamasini yeching, bunda ± manfiy. -200 dan 4i\sqrt{1910} ni ayirish.

$$x=\frac{-4\sqrt{1910}i-200}{126}$$

-200-4i\sqrt{1910} ni 126 ga bo'lish.

$$x=\frac{-2\sqrt{1910}i-100}{63}$$

Tenglama yechildi.

$$x=\frac{-100+2\sqrt{1910}i}{63} x=\frac{-2\sqrt{1910}i-100}{63}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$25\left(16+8x+x^{2}\right)+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

25 ga 16+8x+x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400+200x+25x^{2}+7\left(5-x\right)\left(5+x\right)=295-45x^{2}$$

7 ga 5-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$400+200x+25x^{2}+\left(35-7x\right)\left(5+x\right)=295-45x^{2}$$

35-7x ga 5+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$400+200x+25x^{2}+175-7x^{2}=295-45x^{2}$$

575 olish uchun 400 va 175'ni qo'shing.

$$575+200x+25x^{2}-7x^{2}=295-45x^{2}$$

18x^{2} ni olish uchun 25x^{2} va -7x^{2} ni birlashtirish.

$$575+200x+18x^{2}=295-45x^{2}$$

45x^{2} ni ikki tarafga qo’shing.

$$575+200x+18x^{2}+45x^{2}=295$$

63x^{2} ni olish uchun 18x^{2} va 45x^{2} ni birlashtirish.

$$575+200x+63x^{2}=295$$

Ikkala tarafdan 575 ni ayirish.

$$200x+63x^{2}=295-575$$

-280 olish uchun 295 dan 575 ni ayirish.

$$200x+63x^{2}=-280$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$63x^{2}+200x=-280$$

Ikki tarafini 63 ga bo‘ling.

$$\frac{63x^{2}+200x}{63}=-\frac{280}{63}$$

63 ga bo'lish 63 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{200}{63}x=-\frac{280}{63}$$

\frac{-280}{63} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{200}{63}x=-\frac{40}{9}$$

\frac{200}{63} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{100}{63} olish uchun. Keyin, \frac{100}{63} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{200}{63}x+\left(\frac{100}{63}\right)^{2}=-\frac{40}{9}+\left(\frac{100}{63}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{100}{63} kvadratini chiqarish.

$$x^{2}+\frac{200}{63}x+\frac{10000}{3969}=-\frac{40}{9}+\frac{10000}{3969}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{40}{9} ni \frac{10000}{3969} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{200}{63}x+\frac{10000}{3969}=-\frac{7640}{3969}$$

x^{2}+\frac{200}{63}x+\frac{10000}{3969} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{100}{63}\right)^{2}=-\frac{7640}{3969}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{100}{63}\right)^{2}}=\sqrt{-\frac{7640}{3969}}$$

Qisqartirish.

$$x+\frac{100}{63}=\frac{2\sqrt{1910}i}{63} x+\frac{100}{63}=-\frac{2\sqrt{1910}i}{63}$$

Tenglamaning ikkala tarafidan \frac{100}{63} ni ayirish.

$$x=\frac{-100+2\sqrt{1910}i}{63} x=\frac{-2\sqrt{1910}i-100}{63}$$