20/x-5=60/x+5+1 eching

$$\frac{ 20 }{ x-5 } = \frac{ 60 }{ x+5 } +1$$

x uchun yechish

$$x=5\sqrt{33}-20\approx 8,722813233$$

$$x=-5\sqrt{33}-20\approx -48,722813233$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -5,5 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-5\right)\left(x+5\right) ga, x-5,x+5 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

x+5 ga 20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

x-5 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Hisoblang: \left(x-5\right)\left(x+5\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 5 kvadratini chiqarish.

$$20x+100=60x-300+x^{2}-25$$

-325 olish uchun -300 dan 25 ni ayirish.

$$20x+100=60x-325+x^{2}$$

Ikkala tarafdan 60x ni ayirish.

$$20x+100-60x=-325+x^{2}$$

-40x ni olish uchun 20x va -60x ni birlashtirish.

$$-40x+100=-325+x^{2}$$

Ikkala tarafdan -325 ni ayirish.

$$-40x+100-\left(-325\right)=x^{2}$$

-325 ning teskarisi 325 ga teng.

$$-40x+100+325=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-40x+100+325-x^{2}=0$$

425 olish uchun 100 va 325'ni qo'shing.

$$-40x+425-x^{2}=0$$

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.

$$-x^{2}-40x+425=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -40 ni b va 425 ni c bilan almashtiring.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

-40 kvadratini chiqarish.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\left(-1\right)\times 425}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600+4\times 425}}{2\left(-1\right)}$$

4 ni 425 marotabaga ko'paytirish.

$$x=\frac{-\left(-40\right)±\sqrt{1600+1700}}{2\left(-1\right)}$$

1600 ni 1700 ga qo'shish.

$$x=\frac{-\left(-40\right)±\sqrt{3300}}{2\left(-1\right)}$$

3300 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-40\right)±10\sqrt{33}}{2\left(-1\right)}$$

-40 ning teskarisi 40 ga teng.

$$x=\frac{40±10\sqrt{33}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{40±10\sqrt{33}}{-2}$$

x=\frac{40±10\sqrt{33}}{-2} tenglamasini yeching, bunda ± musbat. 40 ni 10\sqrt{33} ga qo'shish.

$$x=\frac{10\sqrt{33}+40}{-2}$$

40+10\sqrt{33} ni -2 ga bo'lish.

$$x=-5\sqrt{33}-20$$

x=\frac{40±10\sqrt{33}}{-2} tenglamasini yeching, bunda ± manfiy. 40 dan 10\sqrt{33} ni ayirish.

$$x=\frac{40-10\sqrt{33}}{-2}$$

40-10\sqrt{33} ni -2 ga bo'lish.

$$x=5\sqrt{33}-20$$

Tenglama yechildi.

$$x=-5\sqrt{33}-20 x=5\sqrt{33}-20$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -5,5 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-5\right)\left(x+5\right) ga, x-5,x+5 ning eng kichik karralisiga ko‘paytiring.

$$\left(x+5\right)\times 20=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

x+5 ga 20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$20x+100=\left(x-5\right)\times 60+\left(x-5\right)\left(x+5\right)$$

x-5 ga 60 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$20x+100=60x-300+\left(x-5\right)\left(x+5\right)$$

Hisoblang: \left(x-5\right)\left(x+5\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 5 kvadratini chiqarish.

$$20x+100=60x-300+x^{2}-25$$

-325 olish uchun -300 dan 25 ni ayirish.

$$20x+100=60x-325+x^{2}$$

Ikkala tarafdan 60x ni ayirish.

$$20x+100-60x=-325+x^{2}$$

-40x ni olish uchun 20x va -60x ni birlashtirish.

$$-40x+100=-325+x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-40x+100-x^{2}=-325$$

Ikkala tarafdan 100 ni ayirish.

$$-40x-x^{2}=-325-100$$

-425 olish uchun -325 dan 100 ni ayirish.

$$-40x-x^{2}=-425$$

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

$$-x^{2}-40x=-425$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-40x}{-1}=-\frac{425}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{40}{-1}\right)x=-\frac{425}{-1}$$

-40 ni -1 ga bo'lish.

$$x^{2}+40x=-\frac{425}{-1}$$

-425 ni -1 ga bo'lish.

$$x^{2}+40x=425$$

40 ni bo‘lish, x shartining koeffitsienti, 2 ga 20 olish uchun. Keyin, 20 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+40x+20^{2}=425+20^{2}$$

20 kvadratini chiqarish.

$$x^{2}+40x+400=425+400$$

425 ni 400 ga qo'shish.

$$x^{2}+40x+400=825$$

x^{2}+40x+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.

$$\left(x+20\right)^{2}=825$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+20\right)^{2}}=\sqrt{825}$$

Qisqartirish.

$$x+20=5\sqrt{33} x+20=-5\sqrt{33}$$

Tenglamaning ikkala tarafidan 20 ni ayirish.

$$x=5\sqrt{33}-20 x=-5\sqrt{33}-20$$