3x+2/9-x^2=5x+1/x+3-1/3-x eching

$$\frac { 3 x + 2 } { 9 - x ^ { 2 } } = \frac { 5 x + 1 } { x + 3 } - \frac { 1 } { 3 - x }$$

x uchun yechish

$$x=\frac{\sqrt{15}}{5}+1\approx 1,774596669$$

$$x=-\frac{\sqrt{15}}{5}+1\approx 0,225403331$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x+3\right) ga, 9-x^{2},x+3,3-x ning eng kichik karralisiga ko‘paytiring.

$$-\left(3x+2\right)=\left(x-3\right)\left(5x+1\right)+3+x$$

3x+2 teskarisini topish uchun har birining teskarisini toping.

$$-3x-2=\left(x-3\right)\left(5x+1\right)+3+x$$

x-3 ga 5x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-3x-2=5x^{2}-14x-3+3+x$$

0 olish uchun -3 va 3'ni qo'shing.

$$-3x-2=5x^{2}-14x+x$$

-13x ni olish uchun -14x va x ni birlashtirish.

$$-3x-2=5x^{2}-13x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$-3x-2-5x^{2}=-13x$$

13x ni ikki tarafga qo’shing.

$$-3x-2-5x^{2}+13x=0$$

10x ni olish uchun -3x va 13x ni birlashtirish.

$$10x-2-5x^{2}=0$$

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.

$$-5x^{2}+10x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 10 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-5\right)\left(-2\right)}}{2\left(-5\right)}$$

10 kvadratini chiqarish.

$$x=\frac{-10±\sqrt{100-4\left(-5\right)\left(-2\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100+20\left(-2\right)}}{2\left(-5\right)}$$

20 ni -2 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100-40}}{2\left(-5\right)}$$

100 ni -40 ga qo'shish.

$$x=\frac{-10±\sqrt{60}}{2\left(-5\right)}$$

60 ning kvadrat ildizini chiqarish.

$$x=\frac{-10±2\sqrt{15}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{-10±2\sqrt{15}}{-10}$$

x=\frac{-10±2\sqrt{15}}{-10} tenglamasini yeching, bunda ± musbat. -10 ni 2\sqrt{15} ga qo'shish.

$$x=\frac{2\sqrt{15}-10}{-10}$$

-10+2\sqrt{15} ni -10 ga bo'lish.

$$x=-\frac{\sqrt{15}}{5}+1$$

x=\frac{-10±2\sqrt{15}}{-10} tenglamasini yeching, bunda ± manfiy. -10 dan 2\sqrt{15} ni ayirish.

$$x=\frac{-2\sqrt{15}-10}{-10}$$

-10-2\sqrt{15} ni -10 ga bo'lish.

$$x=\frac{\sqrt{15}}{5}+1$$

Tenglama yechildi.

$$x=-\frac{\sqrt{15}}{5}+1 x=\frac{\sqrt{15}}{5}+1$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x+3\right) ga, 9-x^{2},x+3,3-x ning eng kichik karralisiga ko‘paytiring.

$$-\left(3x+2\right)=\left(x-3\right)\left(5x+1\right)+3+x$$

3x+2 teskarisini topish uchun har birining teskarisini toping.

$$-3x-2=\left(x-3\right)\left(5x+1\right)+3+x$$

x-3 ga 5x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-3x-2=5x^{2}-14x-3+3+x$$

0 olish uchun -3 va 3'ni qo'shing.

$$-3x-2=5x^{2}-14x+x$$

-13x ni olish uchun -14x va x ni birlashtirish.

$$-3x-2=5x^{2}-13x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$-3x-2-5x^{2}=-13x$$

13x ni ikki tarafga qo’shing.

$$-3x-2-5x^{2}+13x=0$$

10x ni olish uchun -3x va 13x ni birlashtirish.

$$10x-2-5x^{2}=0$$

2 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$10x-5x^{2}=2$$

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

$$-5x^{2}+10x=2$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}+10x}{-5}=\frac{2}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{10}{-5}x=\frac{2}{-5}$$

10 ni -5 ga bo'lish.

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

2 ni -5 ga bo'lish.

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

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=-\frac{2}{5}+1$$

-\frac{2}{5} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{3}{5}$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=\frac{3}{5}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{3}{5}}$$

Qisqartirish.

$$x-1=\frac{\sqrt{15}}{5} x-1=-\frac{\sqrt{15}}{5}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{15}}{5}+1 x=-\frac{\sqrt{15}}{5}+1$$