-18a^2-102a+168 eching

$$- 18 a ^ { 2 } - 102 a + 168$$

Omil

$$6\left(-a-7\right)\left(3a-4\right)$$

Yechim qadamlari

6 omili.

$$6\left(-3a^{2}-17a+28\right)$$

Hisoblang: -3a^{2}-17a+28. Ifodani guruhlash orqali faktorlang. Avvalo, ifoda -3a^{2}+pa+qa+28 sifatida qayta yozilishi kerak. p va q ni topish uchun yechiladigan tizimni sozlang.

$$p+q=-17 pq=-3\times 28=-84$$

pq manfiy boʻlganda, p va q da qarama-qarshi belgilar bor. p+q manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -84-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Har bir juftlik yigʻindisini hisoblang.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

Yechim – -17 yigʻindisini beruvchi juftlik.

$$p=4 q=-21$$

-3a^{2}-17a+28 ni \left(-3a^{2}+4a\right)+\left(-21a+28\right) sifatida qaytadan yozish.

$$\left(-3a^{2}+4a\right)+\left(-21a+28\right)$$

Birinchi guruhda -a ni va ikkinchi guruhda -7 ni faktordan chiqaring.

$$-a\left(3a-4\right)-7\left(3a-4\right)$$

Distributiv funktsiyasidan foydalangan holda 3a-4 umumiy terminini chiqaring.

$$\left(3a-4\right)\left(-a-7\right)$$

Toʻliq ajratilgan ifodani qaytadan yozing.

$$6\left(3a-4\right)\left(-a-7\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$-18a^{2}-102a+168=0$$

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.

$$a=\frac{-\left(-102\right)±\sqrt{\left(-102\right)^{2}-4\left(-18\right)\times 168}}{2\left(-18\right)}$$

-102 kvadratini chiqarish.

$$a=\frac{-\left(-102\right)±\sqrt{10404-4\left(-18\right)\times 168}}{2\left(-18\right)}$$

-4 ni -18 marotabaga ko'paytirish.

$$a=\frac{-\left(-102\right)±\sqrt{10404+72\times 168}}{2\left(-18\right)}$$

72 ni 168 marotabaga ko'paytirish.

$$a=\frac{-\left(-102\right)±\sqrt{10404+12096}}{2\left(-18\right)}$$

10404 ni 12096 ga qo'shish.

$$a=\frac{-\left(-102\right)±\sqrt{22500}}{2\left(-18\right)}$$

22500 ning kvadrat ildizini chiqarish.

$$a=\frac{-\left(-102\right)±150}{2\left(-18\right)}$$

-102 ning teskarisi 102 ga teng.

$$a=\frac{102±150}{2\left(-18\right)}$$

2 ni -18 marotabaga ko'paytirish.

$$a=\frac{102±150}{-36}$$

a=\frac{102±150}{-36} tenglamasini yeching, bunda ± musbat. 102 ni 150 ga qo'shish.

$$a=\frac{252}{-36}$$

252 ni -36 ga bo'lish.

$$a=-7$$

a=\frac{102±150}{-36} tenglamasini yeching, bunda ± manfiy. 102 dan 150 ni ayirish.

$$a=-\frac{48}{-36}$$

\frac{-48}{-36} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=\frac{4}{3}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun -7 ga va x_{2} uchun \frac{4}{3} ga bo‘ling.

$$-18a^{2}-102a+168=-18\left(a-\left(-7\right)\right)\left(a-\frac{4}{3}\right)$$

p-\left(-q\right) shaklining barcha amallarigani p+q ga soddalashtiring.

$$-18a^{2}-102a+168=-18\left(a+7\right)\left(a-\frac{4}{3}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{4}{3} ni a dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$-18a^{2}-102a+168=-18\left(a+7\right)\times \frac{-3a+4}{-3}$$

-18 va 3 ichida eng katta umumiy 3 faktorini bekor qiling.

$$-18a^{2}-102a+168=6\left(a+7\right)\left(-3a+4\right)$$

Baholash

$$168-102a-18a^{2}$$