20x^2-30x-200 eching

$$20 x ^ { 2 } - 30 x - 200$$

Omil

$$10\left(x-4\right)\left(2x+5\right)$$

Yechim qadamlari

10 omili.

$$10\left(2x^{2}-3x-20\right)$$

Hisoblang: 2x^{2}-3x-20. Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 2x^{2}+ax+bx-20 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=2\left(-20\right)=-40$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -40-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-40 2,-20 4,-10 5,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

Yechim – -3 yigʻindisini beruvchi juftlik.

$$a=-8 b=5$$

2x^{2}-3x-20 ni \left(2x^{2}-8x\right)+\left(5x-20\right) sifatida qaytadan yozish.

$$\left(2x^{2}-8x\right)+\left(5x-20\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$2x\left(x-4\right)+5\left(x-4\right)$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(2x+5\right)$$

Toʻliq ajratilgan ifodani qaytadan yozing.

$$10\left(x-4\right)\left(2x+5\right)$$

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.

$$20x^{2}-30x-200=0$$

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.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 20\left(-200\right)}}{2\times 20}$$

-30 kvadratini chiqarish.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 20\left(-200\right)}}{2\times 20}$$

-4 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-80\left(-200\right)}}{2\times 20}$$

-80 ni -200 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900+16000}}{2\times 20}$$

900 ni 16000 ga qo'shish.

$$x=\frac{-\left(-30\right)±\sqrt{16900}}{2\times 20}$$

16900 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-30\right)±130}{2\times 20}$$

-30 ning teskarisi 30 ga teng.

$$x=\frac{30±130}{2\times 20}$$

2 ni 20 marotabaga ko'paytirish.

$$x=\frac{30±130}{40}$$

x=\frac{30±130}{40} tenglamasini yeching, bunda ± musbat. 30 ni 130 ga qo'shish.

$$x=\frac{160}{40}$$

160 ni 40 ga bo'lish.

$$x=4$$

x=\frac{30±130}{40} tenglamasini yeching, bunda ± manfiy. 30 dan 130 ni ayirish.

$$x=-\frac{100}{40}$$

\frac{-100}{40} ulushini 20 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun 4 ga va x_{2} uchun -\frac{5}{2} ga bo‘ling.

$$20x^{2}-30x-200=20\left(x-4\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$20x^{2}-30x-200=20\left(x-4\right)\left(x+\frac{5}{2}\right)$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni x ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$20x^{2}-30x-200=20\left(x-4\right)\times \frac{2x+5}{2}$$

20 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$20x^{2}-30x-200=10\left(x-4\right)\left(2x+5\right)$$

Baholash

$$10\left(x-4\right)\left(2x+5\right)$$