2x-2x^2/x-2=12 eching

$$\frac { 2 x - 2 x ^ { 2 } } { x - 2 } = 12$$

x uchun yechish

$$x = \frac{\sqrt{73} - 5}{2} \approx 1,772001873$$

$$x=\frac{-\sqrt{73}-5}{2}\approx -6,772001873$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$2x-2x^{2}=12\left(x-2\right)$$

12 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x-2x^{2}=12x-24$$

Ikkala tarafdan 12x ni ayirish.

$$2x-2x^{2}-12x=-24$$

-10x ni olish uchun 2x va -12x ni birlashtirish.

$$-10x-2x^{2}=-24$$

24 ni ikki tarafga qo’shing.

$$-10x-2x^{2}+24=0$$

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.

$$-2x^{2}-10x+24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, -10 ni b va 24 ni c bilan almashtiring.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

-10 kvadratini chiqarish.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100+8\times 24}}{2\left(-2\right)}$$

8 ni 24 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100+192}}{2\left(-2\right)}$$

100 ni 192 ga qo'shish.

$$x=\frac{-\left(-10\right)±\sqrt{292}}{2\left(-2\right)}$$

292 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-10\right)±2\sqrt{73}}{2\left(-2\right)}$$

-10 ning teskarisi 10 ga teng.

$$x=\frac{10±2\sqrt{73}}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{10±2\sqrt{73}}{-4}$$

x=\frac{10±2\sqrt{73}}{-4} tenglamasini yeching, bunda ± musbat. 10 ni 2\sqrt{73} ga qo'shish.

$$x=\frac{2\sqrt{73}+10}{-4}$$

10+2\sqrt{73} ni -4 ga bo'lish.

$$x=\frac{-\sqrt{73}-5}{2}$$

x=\frac{10±2\sqrt{73}}{-4} tenglamasini yeching, bunda ± manfiy. 10 dan 2\sqrt{73} ni ayirish.

$$x=\frac{10-2\sqrt{73}}{-4}$$

10-2\sqrt{73} ni -4 ga bo'lish.

$$x=\frac{\sqrt{73}-5}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{73}-5}{2} x=\frac{\sqrt{73}-5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 2 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x-2 ga ko'paytirish.

$$2x-2x^{2}=12\left(x-2\right)$$

12 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x-2x^{2}=12x-24$$

Ikkala tarafdan 12x ni ayirish.

$$2x-2x^{2}-12x=-24$$

-10x ni olish uchun 2x va -12x ni birlashtirish.

$$-10x-2x^{2}=-24$$

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

$$-2x^{2}-10x=-24$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}-10x}{-2}=-\frac{24}{-2}$$

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{10}{-2}\right)x=-\frac{24}{-2}$$

-10 ni -2 ga bo'lish.

$$x^{2}+5x=-\frac{24}{-2}$$

-24 ni -2 ga bo'lish.

$$x^{2}+5x=12$$

5 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{2} olish uchun. Keyin, \frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=12+\left(\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$x^{2}+5x+\frac{25}{4}=12+\frac{25}{4}$$

12 ni \frac{25}{4} ga qo'shish.

$$x^{2}+5x+\frac{25}{4}=\frac{73}{4}$$

x^{2}+5x+\frac{25}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{73}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{73}{4}}$$

Qisqartirish.

$$x+\frac{5}{2}=\frac{\sqrt{73}}{2} x+\frac{5}{2}=-\frac{\sqrt{73}}{2}$$

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$x=\frac{\sqrt{73}-5}{2} x=\frac{-\sqrt{73}-5}{2}$$