(40-4t)^2+(30-3t)^2=30 eching

$$( 40 - 4 t ) ^ { 2 } + ( 30 - 3 t ) ^ { 2 } = 30$$

t uchun yechish

$$t=\frac{\sqrt{30}}{5}+10\approx 11,095445115$$

$$t=-\frac{\sqrt{30}}{5}+10\approx 8,904554885$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(40-4t\right)^{2} kengaytirilishi uchun ishlating.

$$1600-320t+16t^{2}+\left(30-3t\right)^{2}=30$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(30-3t\right)^{2} kengaytirilishi uchun ishlating.

$$1600-320t+16t^{2}+900-180t+9t^{2}=30$$

2500 olish uchun 1600 va 900'ni qo'shing.

$$2500-320t+16t^{2}-180t+9t^{2}=30$$

-500t ni olish uchun -320t va -180t ni birlashtirish.

$$2500-500t+16t^{2}+9t^{2}=30$$

25t^{2} ni olish uchun 16t^{2} va 9t^{2} ni birlashtirish.

$$2500-500t+25t^{2}=30$$

Ikkala tarafdan 30 ni ayirish.

$$2500-500t+25t^{2}-30=0$$

2470 olish uchun 2500 dan 30 ni ayirish.

$$2470-500t+25t^{2}=0$$

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.

$$25t^{2}-500t+2470=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 25 ni a, -500 ni b va 2470 ni c bilan almashtiring.

$$t=\frac{-\left(-500\right)±\sqrt{\left(-500\right)^{2}-4\times 25\times 2470}}{2\times 25}$$

-500 kvadratini chiqarish.

$$t=\frac{-\left(-500\right)±\sqrt{250000-4\times 25\times 2470}}{2\times 25}$$

-4 ni 25 marotabaga ko'paytirish.

$$t=\frac{-\left(-500\right)±\sqrt{250000-100\times 2470}}{2\times 25}$$

-100 ni 2470 marotabaga ko'paytirish.

$$t=\frac{-\left(-500\right)±\sqrt{250000-247000}}{2\times 25}$$

250000 ni -247000 ga qo'shish.

$$t=\frac{-\left(-500\right)±\sqrt{3000}}{2\times 25}$$

3000 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-500\right)±10\sqrt{30}}{2\times 25}$$

-500 ning teskarisi 500 ga teng.

$$t=\frac{500±10\sqrt{30}}{2\times 25}$$

2 ni 25 marotabaga ko'paytirish.

$$t=\frac{500±10\sqrt{30}}{50}$$

t=\frac{500±10\sqrt{30}}{50} tenglamasini yeching, bunda ± musbat. 500 ni 10\sqrt{30} ga qo'shish.

$$t=\frac{10\sqrt{30}+500}{50}$$

500+10\sqrt{30} ni 50 ga bo'lish.

$$t=\frac{\sqrt{30}}{5}+10$$

t=\frac{500±10\sqrt{30}}{50} tenglamasini yeching, bunda ± manfiy. 500 dan 10\sqrt{30} ni ayirish.

$$t=\frac{500-10\sqrt{30}}{50}$$

500-10\sqrt{30} ni 50 ga bo'lish.

$$t=-\frac{\sqrt{30}}{5}+10$$

Tenglama yechildi.

$$t=\frac{\sqrt{30}}{5}+10 t=-\frac{\sqrt{30}}{5}+10$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(40-4t\right)^{2} kengaytirilishi uchun ishlating.

$$1600-320t+16t^{2}+\left(30-3t\right)^{2}=30$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(30-3t\right)^{2} kengaytirilishi uchun ishlating.

$$1600-320t+16t^{2}+900-180t+9t^{2}=30$$

2500 olish uchun 1600 va 900'ni qo'shing.

$$2500-320t+16t^{2}-180t+9t^{2}=30$$

-500t ni olish uchun -320t va -180t ni birlashtirish.

$$2500-500t+16t^{2}+9t^{2}=30$$

25t^{2} ni olish uchun 16t^{2} va 9t^{2} ni birlashtirish.

$$2500-500t+25t^{2}=30$$

Ikkala tarafdan 2500 ni ayirish.

$$-500t+25t^{2}=30-2500$$

-2470 olish uchun 30 dan 2500 ni ayirish.

$$-500t+25t^{2}=-2470$$

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

$$25t^{2}-500t=-2470$$

Ikki tarafini 25 ga bo‘ling.

$$\frac{25t^{2}-500t}{25}=-\frac{2470}{25}$$

25 ga bo'lish 25 ga ko'paytirishni bekor qiladi.

$$t^{2}+\left(-\frac{500}{25}\right)t=-\frac{2470}{25}$$

-500 ni 25 ga bo'lish.

$$t^{2}-20t=-\frac{2470}{25}$$

\frac{-2470}{25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-20t=-\frac{494}{5}$$

-20 ni bo‘lish, x shartining koeffitsienti, 2 ga -10 olish uchun. Keyin, -10 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-20t+\left(-10\right)^{2}=-\frac{494}{5}+\left(-10\right)^{2}$$

-10 kvadratini chiqarish.

$$t^{2}-20t+100=-\frac{494}{5}+100$$

-\frac{494}{5} ni 100 ga qo'shish.

$$t^{2}-20t+100=\frac{6}{5}$$

t^{2}-20t+100 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-10\right)^{2}=\frac{6}{5}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-10\right)^{2}}=\sqrt{\frac{6}{5}}$$

Qisqartirish.

$$t-10=\frac{\sqrt{30}}{5} t-10=-\frac{\sqrt{30}}{5}$$

10 ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{30}}{5}+10 t=-\frac{\sqrt{30}}{5}+10$$