-2x^2+2x+9=-5x eching

$$- 2 x ^ { 2 } + 2 x + 9 = - 5 x$$

x uchun yechish

$$x=-1$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4,5$$

Guruhlash orqali faktorlash qadamlari

5x ni ikki tarafga qo’shing.

$$-2x^{2}+2x+9+5x=0$$

7x ni olish uchun 2x va 5x ni birlashtirish.

$$-2x^{2}+7x+9=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx+9 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=7 ab=-2\times 9=-18$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -18-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,18 -2,9 -3,6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+18=17 -2+9=7 -3+6=3$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=9 b=-2$$

-2x^{2}+7x+9 ni \left(-2x^{2}+9x\right)+\left(-2x+9\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+9x\right)+\left(-2x+9\right)$$

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

$$-x\left(2x-9\right)-\left(2x-9\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-9 umumiy terminini chiqaring.

$$\left(2x-9\right)\left(-x-1\right)$$

Tenglamani yechish uchun 2x-9=0 va -x-1=0 ni yeching.

$$x=\frac{9}{2} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

5x ni ikki tarafga qo’shing.

$$-2x^{2}+2x+9+5x=0$$

7x ni olish uchun 2x va 5x ni birlashtirish.

$$-2x^{2}+7x+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 7 ni b va 9 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\times 9}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+8\times 9}}{2\left(-2\right)}$$

8 ni 9 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+72}}{2\left(-2\right)}$$

49 ni 72 ga qo'shish.

$$x=\frac{-7±\sqrt{121}}{2\left(-2\right)}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±11}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±11}{-4}$$

x=\frac{-7±11}{-4} tenglamasini yeching, bunda ± musbat. -7 ni 11 ga qo'shish.

$$x=\frac{4}{-4}$$

4 ni -4 ga bo'lish.

$$x=-1$$

x=\frac{-7±11}{-4} tenglamasini yeching, bunda ± manfiy. -7 dan 11 ni ayirish.

$$x=-\frac{18}{-4}$$

\frac{-18}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{2}$$

Tenglama yechildi.

$$x=-1 x=\frac{9}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

5x ni ikki tarafga qo’shing.

$$-2x^{2}+2x+9+5x=0$$

7x ni olish uchun 2x va 5x ni birlashtirish.

$$-2x^{2}+7x+9=0$$

Ikkala tarafdan 9 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-2x^{2}+7x=-9$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+7x}{-2}=-\frac{9}{-2}$$

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

$$x^{2}+\frac{7}{-2}x=-\frac{9}{-2}$$

7 ni -2 ga bo'lish.

$$x^{2}-\frac{7}{2}x=-\frac{9}{-2}$$

-9 ni -2 ga bo'lish.

$$x^{2}-\frac{7}{2}x=\frac{9}{2}$$

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=\frac{9}{2}+\left(-\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{9}{2}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{2} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{121}{16}$$

x^{2}-\frac{7}{2}x+\frac{49}{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.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{121}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{121}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{11}{4} x-\frac{7}{4}=-\frac{11}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9}{2} x=-1$$