(20t)^2+(90-30t)^2=2500 eching

$$( 20 t ) ^ { 2 } + ( 90 - 30 t ) ^ { 2 } = 2500$$

t uchun yechish

$$t=2$$

$$t = \frac{28}{13} = 2\frac{2}{13} \approx 2,153846154$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(20t\right)^{2} ni kengaytirish.

$$20^{2}t^{2}+\left(90-30t\right)^{2}=2500$$

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$400t^{2}+\left(90-30t\right)^{2}=2500$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(90-30t\right)^{2} kengaytirilishi uchun ishlating.

$$400t^{2}[phone]t+900t^{2}=2500$$

1300t^{2} ni olish uchun 400t^{2} va 900t^{2} ni birlashtirish.

$$1300t^{2}[phone]t=2500$$

Ikkala tarafdan 2500 ni ayirish.

$$1300t^{2}[phone]t-2500=0$$

5600 olish uchun 8100 dan 2500 ni ayirish.

$$1300t^{2}[phone]t=0$$

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.

$$1300t^{2}-5400t+5600=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1300 ni a, -5400 ni b va 5600 ni c bilan almashtiring.

$$t=\frac{-\left(-5400\right)±\sqrt{\left(-5400\right)^{2}-4\times 1300\times 5600}}{2\times 1300}$$

-5400 kvadratini chiqarish.

$$t=\frac{-\left(-5400\right)±\sqrt{29160000-4\times 1300\times 5600}}{2\times 1300}$$

-4 ni 1300 marotabaga ko'paytirish.

$$t=\frac{-\left(-5400\right)±\sqrt{29160000-5200\times 5600}}{2\times 1300}$$

-5200 ni 5600 marotabaga ko'paytirish.

$$t=\frac{-\left(-5400\right)±\sqrt{29160000-29120000}}{2\times 1300}$$

29160000 ni -29120000 ga qo'shish.

$$t=\frac{-\left(-5400\right)±\sqrt{40000}}{2\times 1300}$$

40000 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-5400\right)±200}{2\times 1300}$$

-5400 ning teskarisi 5400 ga teng.

$$t=\frac{5400±200}{2\times 1300}$$

2 ni 1300 marotabaga ko'paytirish.

$$t=\frac{5400±200}{2600}$$

t=\frac{5400±200}{2600} tenglamasini yeching, bunda ± musbat. 5400 ni 200 ga qo'shish.

$$t=\frac{5600}{2600}$$

\frac{5600}{2600} ulushini 200 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t=\frac{28}{13}$$

t=\frac{5400±200}{2600} tenglamasini yeching, bunda ± manfiy. 5400 dan 200 ni ayirish.

$$t=\frac{5200}{2600}$$

5200 ni 2600 ga bo'lish.

$$t=2$$

Tenglama yechildi.

$$t=\frac{28}{13} t=2$$

Kvadratni yakunlash bo'yicha qadamlar

\left(20t\right)^{2} ni kengaytirish.

$$20^{2}t^{2}+\left(90-30t\right)^{2}=2500$$

2 daraja ko‘rsatkichini 20 ga hisoblang va 400 ni qiymatni oling.

$$400t^{2}+\left(90-30t\right)^{2}=2500$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(90-30t\right)^{2} kengaytirilishi uchun ishlating.

$$400t^{2}[phone]t+900t^{2}=2500$$

1300t^{2} ni olish uchun 400t^{2} va 900t^{2} ni birlashtirish.

$$1300t^{2}[phone]t=2500$$

Ikkala tarafdan 8100 ni ayirish.

$$1300t^{2}-5400t=2500-8100$$

-5600 olish uchun 2500 dan 8100 ni ayirish.

$$1300t^{2}-5400t=-5600$$

Ikki tarafini 1300 ga bo‘ling.

$$\frac{1300t^{2}-5400t}{1300}=-\frac{5600}{1300}$$

1300 ga bo'lish 1300 ga ko'paytirishni bekor qiladi.

$$t^{2}+\left(-\frac{5400}{1300}\right)t=-\frac{5600}{1300}$$

\frac{-5400}{1300} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{54}{13}t=-\frac{5600}{1300}$$

\frac{-5600}{1300} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{54}{13}t=-\frac{56}{13}$$

-\frac{54}{13} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{27}{13} olish uchun. Keyin, -\frac{27}{13} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{54}{13}t+\left(-\frac{27}{13}\right)^{2}=-\frac{56}{13}+\left(-\frac{27}{13}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{27}{13} kvadratini chiqarish.

$$t^{2}-\frac{54}{13}t+\frac{729}{169}=-\frac{56}{13}+\frac{729}{169}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{56}{13} ni \frac{729}{169} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{54}{13}t+\frac{729}{169}=\frac{1}{169}$$

t^{2}-\frac{54}{13}t+\frac{729}{169} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{27}{13}\right)^{2}=\frac{1}{169}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{27}{13}\right)^{2}}=\sqrt{\frac{1}{169}}$$

Qisqartirish.

$$t-\frac{27}{13}=\frac{1}{13} t-\frac{27}{13}=-\frac{1}{13}$$

\frac{27}{13} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{28}{13} t=2$$