8x-x(x+3)=2 eching

$$8x-x(x+3)=2$$

x uchun yechish

$$x = \frac{\sqrt{17} + 5}{2} \approx 4,561552813$$

$$x=\frac{5-\sqrt{17}}{2}\approx 0,438447187$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

x^{2}+3x teskarisini topish uchun har birining teskarisini toping.

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

5x ni olish uchun 8x va -3x ni birlashtirish.

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

Ikkala tarafdan 2 ni ayirish.

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

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 5 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+4\left(-2\right)}}{2\left(-1\right)}$$

4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-8}}{2\left(-1\right)}$$

25 ni -8 ga qo'shish.

$$x=\frac{-5±\sqrt{17}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{17}}{-2}$$

x=\frac{-5±\sqrt{17}}{-2} tenglamasini yeching, bunda ± musbat. -5 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}-5}{-2}$$

-5+\sqrt{17} ni -2 ga bo'lish.

$$x=\frac{5-\sqrt{17}}{2}$$

x=\frac{-5±\sqrt{17}}{-2} tenglamasini yeching, bunda ± manfiy. -5 dan \sqrt{17} ni ayirish.

$$x=\frac{-\sqrt{17}-5}{-2}$$

-5-\sqrt{17} ni -2 ga bo'lish.

$$x=\frac{\sqrt{17}+5}{2}$$

Tenglama yechildi.

$$x=\frac{5-\sqrt{17}}{2} x=\frac{\sqrt{17}+5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

x^{2}+3x teskarisini topish uchun har birining teskarisini toping.

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

5x ni olish uchun 8x va -3x ni birlashtirish.

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

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

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

5 ni -1 ga bo'lish.

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

2 ni -1 ga bo'lish.

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

-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.

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

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

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

-2 ni \frac{25}{4} ga qo'shish.

$$x^{2}-5x+\frac{25}{4}=\frac{17}{4}$$

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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{17}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{17}}{2} x-\frac{5}{2}=-\frac{\sqrt{17}}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{17}+5}{2} x=\frac{5-\sqrt{17}}{2}$$