-3r^2+90r=93 eching

$$- 3 r ^ { 2 } + 90 r = 93$$

r uchun yechish

$$r=\sqrt{194}+15\approx 28,928388277$$

$$r=15-\sqrt{194}\approx 1,071611723$$

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.

$$-3r^{2}+90r=93$$

Tenglamaning ikkala tarafidan 93 ni ayirish.

$$-3r^{2}+90r-93=93-93$$

O‘zidan 93 ayirilsa 0 qoladi.

$$-3r^{2}+90r-93=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 90 ni b va -93 ni c bilan almashtiring.

$$r=\frac{-90±\sqrt{90^{2}-4\left(-3\right)\left(-93\right)}}{2\left(-3\right)}$$

90 kvadratini chiqarish.

$$r=\frac{-90±\sqrt{8100-4\left(-3\right)\left(-93\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$r=\frac{-90±\sqrt{8100+12\left(-93\right)}}{2\left(-3\right)}$$

12 ni -93 marotabaga ko'paytirish.

$$r=\frac{-90±\sqrt{8100-1116}}{2\left(-3\right)}$$

8100 ni -1116 ga qo'shish.

$$r=\frac{-90±\sqrt{6984}}{2\left(-3\right)}$$

6984 ning kvadrat ildizini chiqarish.

$$r=\frac{-90±6\sqrt{194}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$r=\frac{-90±6\sqrt{194}}{-6}$$

r=\frac{-90±6\sqrt{194}}{-6} tenglamasini yeching, bunda ± musbat. -90 ni 6\sqrt{194} ga qo'shish.

$$r=\frac{6\sqrt{194}-90}{-6}$$

-90+6\sqrt{194} ni -6 ga bo'lish.

$$r=15-\sqrt{194}$$

r=\frac{-90±6\sqrt{194}}{-6} tenglamasini yeching, bunda ± manfiy. -90 dan 6\sqrt{194} ni ayirish.

$$r=\frac{-6\sqrt{194}-90}{-6}$$

-90-6\sqrt{194} ni -6 ga bo'lish.

$$r=\sqrt{194}+15$$

Tenglama yechildi.

$$r=15-\sqrt{194} r=\sqrt{194}+15$$

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.

$$-3r^{2}+90r=93$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3r^{2}+90r}{-3}=\frac{93}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$r^{2}+\frac{90}{-3}r=\frac{93}{-3}$$

90 ni -3 ga bo'lish.

$$r^{2}-30r=\frac{93}{-3}$$

93 ni -3 ga bo'lish.

$$r^{2}-30r=-31$$

-30 ni bo‘lish, x shartining koeffitsienti, 2 ga -15 olish uchun. Keyin, -15 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$r^{2}-30r+\left(-15\right)^{2}=-31+\left(-15\right)^{2}$$

-15 kvadratini chiqarish.

$$r^{2}-30r+225=-31+225$$

-31 ni 225 ga qo'shish.

$$r^{2}-30r+225=194$$

r^{2}-30r+225 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(r-15\right)^{2}=194$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(r-15\right)^{2}}=\sqrt{194}$$

Qisqartirish.

$$r-15=\sqrt{194} r-15=-\sqrt{194}$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$r=\sqrt{194}+15 r=15-\sqrt{194}$$