30t+2t^2=300 eching

$$30 t + 2 t ^ { 2 } = 300$$

t uchun yechish

$$t = \frac{5 \sqrt{33} - 15}{2} \approx 6,861406616$$

$$t=\frac{-5\sqrt{33}-15}{2}\approx -21,861406616$$

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.

$$2t^{2}+30t=300$$

Tenglamaning ikkala tarafidan 300 ni ayirish.

$$2t^{2}+30t-300=300-300$$

O‘zidan 300 ayirilsa 0 qoladi.

$$2t^{2}+30t-300=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 30 ni b va -300 ni c bilan almashtiring.

$$t=\frac{-30±\sqrt{30^{2}-4\times 2\left(-300\right)}}{2\times 2}$$

30 kvadratini chiqarish.

$$t=\frac{-30±\sqrt{900-4\times 2\left(-300\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$t=\frac{-30±\sqrt{900-8\left(-300\right)}}{2\times 2}$$

-8 ni -300 marotabaga ko'paytirish.

$$t=\frac{-30±\sqrt{900+2400}}{2\times 2}$$

900 ni 2400 ga qo'shish.

$$t=\frac{-30±\sqrt{3300}}{2\times 2}$$

3300 ning kvadrat ildizini chiqarish.

$$t=\frac{-30±10\sqrt{33}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$t=\frac{-30±10\sqrt{33}}{4}$$

t=\frac{-30±10\sqrt{33}}{4} tenglamasini yeching, bunda ± musbat. -30 ni 10\sqrt{33} ga qo'shish.

$$t=\frac{10\sqrt{33}-30}{4}$$

-30+10\sqrt{33} ni 4 ga bo'lish.

$$t=\frac{5\sqrt{33}-15}{2}$$

t=\frac{-30±10\sqrt{33}}{4} tenglamasini yeching, bunda ± manfiy. -30 dan 10\sqrt{33} ni ayirish.

$$t=\frac{-10\sqrt{33}-30}{4}$$

-30-10\sqrt{33} ni 4 ga bo'lish.

$$t=\frac{-5\sqrt{33}-15}{2}$$

Tenglama yechildi.

$$t=\frac{5\sqrt{33}-15}{2} t=\frac{-5\sqrt{33}-15}{2}$$

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.

$$2t^{2}+30t=300$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2t^{2}+30t}{2}=\frac{300}{2}$$

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

$$t^{2}+\frac{30}{2}t=\frac{300}{2}$$

30 ni 2 ga bo'lish.

$$t^{2}+15t=\frac{300}{2}$$

300 ni 2 ga bo'lish.

$$t^{2}+15t=150$$

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

$$t^{2}+15t+\left(\frac{15}{2}\right)^{2}=150+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$t^{2}+15t+\frac{225}{4}=150+\frac{225}{4}$$

150 ni \frac{225}{4} ga qo'shish.

$$t^{2}+15t+\frac{225}{4}=\frac{825}{4}$$

t^{2}+15t+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+\frac{15}{2}\right)^{2}=\frac{825}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+\frac{15}{2}\right)^{2}}=\sqrt{\frac{825}{4}}$$

Qisqartirish.

$$t+\frac{15}{2}=\frac{5\sqrt{33}}{2} t+\frac{15}{2}=-\frac{5\sqrt{33}}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$t=\frac{5\sqrt{33}-15}{2} t=\frac{-5\sqrt{33}-15}{2}$$