(x)=3x^2-7x+2 eching

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

x uchun yechish

$$x = \frac{\sqrt{10} + 4}{3} \approx 2,387425887$$

$$x=\frac{4-\sqrt{10}}{3}\approx 0,27924078$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-7x+2$$

7x ni ikki tarafga qo’shing.

$$x-3x^{2}+7x=2$$

8x ni olish uchun x va 7x ni birlashtirish.

$$8x-3x^{2}=2$$

Ikkala tarafdan 2 ni ayirish.

$$8x-3x^{2}-2=0$$

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.

$$-3x^{2}+8x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 8 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64+12\left(-2\right)}}{2\left(-3\right)}$$

12 ni -2 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64-24}}{2\left(-3\right)}$$

64 ni -24 ga qo'shish.

$$x=\frac{-8±\sqrt{40}}{2\left(-3\right)}$$

40 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±2\sqrt{10}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-8±2\sqrt{10}}{-6}$$

x=\frac{-8±2\sqrt{10}}{-6} tenglamasini yeching, bunda ± musbat. -8 ni 2\sqrt{10} ga qo'shish.

$$x=\frac{2\sqrt{10}-8}{-6}$$

-8+2\sqrt{10} ni -6 ga bo'lish.

$$x=\frac{4-\sqrt{10}}{3}$$

x=\frac{-8±2\sqrt{10}}{-6} tenglamasini yeching, bunda ± manfiy. -8 dan 2\sqrt{10} ni ayirish.

$$x=\frac{-2\sqrt{10}-8}{-6}$$

-8-2\sqrt{10} ni -6 ga bo'lish.

$$x=\frac{\sqrt{10}+4}{3}$$

Tenglama yechildi.

$$x=\frac{4-\sqrt{10}}{3} x=\frac{\sqrt{10}+4}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-7x+2$$

7x ni ikki tarafga qo’shing.

$$x-3x^{2}+7x=2$$

8x ni olish uchun x va 7x ni birlashtirish.

$$8x-3x^{2}=2$$

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

$$-3x^{2}+8x=2$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+8x}{-3}=\frac{2}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{8}{-3}x=\frac{2}{-3}$$

8 ni -3 ga bo'lish.

$$x^{2}-\frac{8}{3}x=\frac{2}{-3}$$

2 ni -3 ga bo'lish.

$$x^{2}-\frac{8}{3}x=-\frac{2}{3}$$

-\frac{8}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{3} olish uchun. Keyin, -\frac{4}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{2}{3}+\left(-\frac{4}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{3} kvadratini chiqarish.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{2}{3}+\frac{16}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{3} ni \frac{16}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{10}{9}$$

x^{2}-\frac{8}{3}x+\frac{16}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{10}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{10}{9}}$$

Qisqartirish.

$$x-\frac{4}{3}=\frac{\sqrt{10}}{3} x-\frac{4}{3}=-\frac{\sqrt{10}}{3}$$

\frac{4}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{10}+4}{3} x=\frac{4-\sqrt{10}}{3}$$