100=20t+1/2*98t^2 eching

$$100 = 20 t + \frac { 1 } { 2 } \times 98 t ^ { 2 }$$

t uchun yechish

$$t = \frac{50 \sqrt{2} - 10}{49} \approx 1,238993431$$

$$t=\frac{-50\sqrt{2}-10}{49}\approx -1,647156696$$

Kvadrat tenglamalaridan foydalanish qoidalari

49 hosil qilish uchun \frac{1}{2} va 98 ni ko'paytirish.

$$100=20t+49t^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$20t+49t^{2}=100$$

Ikkala tarafdan 100 ni ayirish.

$$20t+49t^{2}-100=0$$

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.

$$49t^{2}+20t-100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, 20 ni b va -100 ni c bilan almashtiring.

$$t=\frac{-20±\sqrt{20^{2}-4\times 49\left(-100\right)}}{2\times 49}$$

20 kvadratini chiqarish.

$$t=\frac{-20±\sqrt{400-4\times 49\left(-100\right)}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$t=\frac{-20±\sqrt{400-196\left(-100\right)}}{2\times 49}$$

-196 ni -100 marotabaga ko'paytirish.

$$t=\frac{-20±\sqrt{400+19600}}{2\times 49}$$

400 ni 19600 ga qo'shish.

$$t=\frac{-20±\sqrt{20000}}{2\times 49}$$

20000 ning kvadrat ildizini chiqarish.

$$t=\frac{-20±100\sqrt{2}}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$t=\frac{-20±100\sqrt{2}}{98}$$

t=\frac{-20±100\sqrt{2}}{98} tenglamasini yeching, bunda ± musbat. -20 ni 100\sqrt{2} ga qo'shish.

$$t=\frac{100\sqrt{2}-20}{98}$$

-20+100\sqrt{2} ni 98 ga bo'lish.

$$t=\frac{50\sqrt{2}-10}{49}$$

t=\frac{-20±100\sqrt{2}}{98} tenglamasini yeching, bunda ± manfiy. -20 dan 100\sqrt{2} ni ayirish.

$$t=\frac{-100\sqrt{2}-20}{98}$$

-20-100\sqrt{2} ni 98 ga bo'lish.

$$t=\frac{-50\sqrt{2}-10}{49}$$

Tenglama yechildi.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$

Kvadratni yakunlash bo'yicha qadamlar

49 hosil qilish uchun \frac{1}{2} va 98 ni ko'paytirish.

$$100=20t+49t^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$20t+49t^{2}=100$$

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

$$49t^{2}+20t=100$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49t^{2}+20t}{49}=\frac{100}{49}$$

49 ga bo'lish 49 ga ko'paytirishni bekor qiladi.

$$t^{2}+\frac{20}{49}t=\frac{100}{49}$$

\frac{20}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{10}{49} olish uchun. Keyin, \frac{10}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}+\frac{20}{49}t+\left(\frac{10}{49}\right)^{2}=\frac{100}{49}+\left(\frac{10}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{10}{49} kvadratini chiqarish.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{100}{49}+\frac{100}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{100}{49} ni \frac{100}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{5000}{2401}$$

t^{2}+\frac{20}{49}t+\frac{100}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+\frac{10}{49}\right)^{2}=\frac{5000}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+\frac{10}{49}\right)^{2}}=\sqrt{\frac{5000}{2401}}$$

Qisqartirish.

$$t+\frac{10}{49}=\frac{50\sqrt{2}}{49} t+\frac{10}{49}=-\frac{50\sqrt{2}}{49}$$

Tenglamaning ikkala tarafidan \frac{10}{49} ni ayirish.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$