5r^2-44r+120=-30 eching

$$5 r ^ { 2 } - 44 r + 120 = - 30$$

r uchun yechish

$$r=\frac{22+\sqrt{266}i}{5}\approx 4,4+3,261901286i$$

$$r=\frac{-\sqrt{266}i+22}{5}\approx 4,4-3,261901286i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$5r^{2}-44r+120=-30$$

30 ni tenglamaning ikkala tarafiga qo'shish.

$$5r^{2}-44r+120-\left(-30\right)=-30-\left(-30\right)$$

O‘zidan -30 ayirilsa 0 qoladi.

$$5r^{2}-44r+120-\left(-30\right)=0$$

120 dan -30 ni ayirish.

$$5r^{2}-44r+150=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -44 ni b va 150 ni c bilan almashtiring.

$$r=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 5\times 150}}{2\times 5}$$

-44 kvadratini chiqarish.

$$r=\frac{-\left(-44\right)±\sqrt{1936-4\times 5\times 150}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$r=\frac{-\left(-44\right)±\sqrt{1936-20\times 150}}{2\times 5}$$

-20 ni 150 marotabaga ko'paytirish.

$$r=\frac{-\left(-44\right)±\sqrt{1936-3000}}{2\times 5}$$

1936 ni -3000 ga qo'shish.

$$r=\frac{-\left(-44\right)±\sqrt{-1064}}{2\times 5}$$

-1064 ning kvadrat ildizini chiqarish.

$$r=\frac{-\left(-44\right)±2\sqrt{266}i}{2\times 5}$$

-44 ning teskarisi 44 ga teng.

$$r=\frac{44±2\sqrt{266}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$r=\frac{44±2\sqrt{266}i}{10}$$

r=\frac{44±2\sqrt{266}i}{10} tenglamasini yeching, bunda ± musbat. 44 ni 2i\sqrt{266} ga qo'shish.

$$r=\frac{44+2\sqrt{266}i}{10}$$

44+2i\sqrt{266} ni 10 ga bo'lish.

$$r=\frac{22+\sqrt{266}i}{5}$$

r=\frac{44±2\sqrt{266}i}{10} tenglamasini yeching, bunda ± manfiy. 44 dan 2i\sqrt{266} ni ayirish.

$$r=\frac{-2\sqrt{266}i+44}{10}$$

44-2i\sqrt{266} ni 10 ga bo'lish.

$$r=\frac{-\sqrt{266}i+22}{5}$$

Tenglama yechildi.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$5r^{2}-44r+120=-30$$

Tenglamaning ikkala tarafidan 120 ni ayirish.

$$5r^{2}-44r+120-120=-30-120$$

O‘zidan 120 ayirilsa 0 qoladi.

$$5r^{2}-44r=-30-120$$

-30 dan 120 ni ayirish.

$$5r^{2}-44r=-150$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5r^{2}-44r}{5}=-\frac{150}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$r^{2}-\frac{44}{5}r=-\frac{150}{5}$$

-150 ni 5 ga bo'lish.

$$r^{2}-\frac{44}{5}r=-30$$

-\frac{44}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{22}{5} olish uchun. Keyin, -\frac{22}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$r^{2}-\frac{44}{5}r+\left(-\frac{22}{5}\right)^{2}=-30+\left(-\frac{22}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{22}{5} kvadratini chiqarish.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-30+\frac{484}{25}$$

-30 ni \frac{484}{25} ga qo'shish.

$$r^{2}-\frac{44}{5}r+\frac{484}{25}=-\frac{266}{25}$$

r^{2}-\frac{44}{5}r+\frac{484}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(r-\frac{22}{5}\right)^{2}=-\frac{266}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(r-\frac{22}{5}\right)^{2}}=\sqrt{-\frac{266}{25}}$$

Qisqartirish.

$$r-\frac{22}{5}=\frac{\sqrt{266}i}{5} r-\frac{22}{5}=-\frac{\sqrt{266}i}{5}$$

\frac{22}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$r=\frac{22+\sqrt{266}i}{5} r=\frac{-\sqrt{266}i+22}{5}$$