-6x^2=3x-3 eching

$$- 6 x ^ { 2 } = 3 x - 3$$

x uchun yechish

$$x=\frac{1}{2}=0,5$$

$$x=-1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 3x ni ayirish.

$$-6x^{2}-3x=-3$$

3 ni ikki tarafga qo’shing.

$$-6x^{2}-3x+3=0$$

Ikki tarafini 3 ga bo‘ling.

$$-2x^{2}-x+1=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx+1 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=-2=-2$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=1 b=-2$$

-2x^{2}-x+1 ni \left(-2x^{2}+x\right)+\left(-2x+1\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+x\right)+\left(-2x+1\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$-x\left(2x-1\right)-\left(2x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-1 umumiy terminini chiqaring.

$$\left(2x-1\right)\left(-x-1\right)$$

Tenglamani yechish uchun 2x-1=0 va -x-1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3x ni ayirish.

$$-6x^{2}-3x=-3$$

3 ni ikki tarafga qo’shing.

$$-6x^{2}-3x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, -3 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-6\right)\times 3}}{2\left(-6\right)}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-6\right)\times 3}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+24\times 3}}{2\left(-6\right)}$$

24 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\left(-6\right)}$$

9 ni 72 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2\left(-6\right)}$$

81 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±9}{2\left(-6\right)}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±9}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{3±9}{-12}$$

x=\frac{3±9}{-12} tenglamasini yeching, bunda ± musbat. 3 ni 9 ga qo'shish.

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

12 ni -12 ga bo'lish.

$$x=-1$$

x=\frac{3±9}{-12} tenglamasini yeching, bunda ± manfiy. 3 dan 9 ni ayirish.

$$x=-\frac{6}{-12}$$

\frac{-6}{-12} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{2}$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 3x ni ayirish.

$$-6x^{2}-3x=-3$$

Ikki tarafini -6 ga bo‘ling.

$$\frac{-6x^{2}-3x}{-6}=-\frac{3}{-6}$$

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{3}{-6}\right)x=-\frac{3}{-6}$$

\frac{-3}{-6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{1}{2}x=-\frac{3}{-6}$$

\frac{-3}{-6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{1}{2}+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}$$

x^{2}+\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{9}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Qisqartirish.

$$x+\frac{1}{4}=\frac{3}{4} x+\frac{1}{4}=-\frac{3}{4}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

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