-49t^2+100t-510204=0 eching

$$- 49 t ^ { 2 } + 100 t - 510204 = 0$$

t uchun yechish

$$t=\frac{-2\sqrt{6249374}i+50}{49}\approx 1,020408163-102,035705994i$$

$$t=\frac{50+2\sqrt{6249374}i}{49}\approx 1,020408163+102,035705994i$$

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.

$$-49t^{2}+100t-510204=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 100 ni b va -510204 ni c bilan almashtiring.

$$t=\frac{-100±\sqrt{100^{2}-4\left(-49\right)\left(-510204\right)}}{2\left(-49\right)}$$

100 kvadratini chiqarish.

$$t=\frac{-100±\sqrt{10000-4\left(-49\right)\left(-510204\right)}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$t=\frac{-100±\sqrt{10000+196\left(-510204\right)}}{2\left(-49\right)}$$

196 ni -510204 marotabaga ko'paytirish.

$$t=\frac{-100±\sqrt{10000-99999984}}{2\left(-49\right)}$$

10000 ni -99999984 ga qo'shish.

$$t=\frac{-100±\sqrt{-99989984}}{2\left(-49\right)}$$

-99989984 ning kvadrat ildizini chiqarish.

$$t=\frac{-100±4\sqrt{6249374}i}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$t=\frac{-100±4\sqrt{6249374}i}{-98}$$

t=\frac{-100±4\sqrt{6249374}i}{-98} tenglamasini yeching, bunda ± musbat. -100 ni 4i\sqrt{6249374} ga qo'shish.

$$t=\frac{-100+4\sqrt{6249374}i}{-98}$$

-100+4i\sqrt{6249374} ni -98 ga bo'lish.

$$t=\frac{-2\sqrt{6249374}i+50}{49}$$

t=\frac{-100±4\sqrt{6249374}i}{-98} tenglamasini yeching, bunda ± manfiy. -100 dan 4i\sqrt{6249374} ni ayirish.

$$t=\frac{-4\sqrt{6249374}i-100}{-98}$$

-100-4i\sqrt{6249374} ni -98 ga bo'lish.

$$t=\frac{50+2\sqrt{6249374}i}{49}$$

Tenglama yechildi.

$$t=\frac{-2\sqrt{6249374}i+50}{49} t=\frac{50+2\sqrt{6249374}i}{49}$$

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.

$$-49t^{2}+100t-510204=0$$

510204 ni tenglamaning ikkala tarafiga qo'shish.

$$-49t^{2}+100t-510204-\left(-510204\right)=-\left(-510204\right)$$

O‘zidan -510204 ayirilsa 0 qoladi.

$$-49t^{2}+100t=-\left(-510204\right)$$

0 dan -510204 ni ayirish.

$$-49t^{2}+100t=510204$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49t^{2}+100t}{-49}=\frac{510204}{-49}$$

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

$$t^{2}+\frac{100}{-49}t=\frac{510204}{-49}$$

100 ni -49 ga bo'lish.

$$t^{2}-\frac{100}{49}t=\frac{510204}{-49}$$

510204 ni -49 ga bo'lish.

$$t^{2}-\frac{100}{49}t=-\frac{510204}{49}$$

-\frac{100}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{50}{49} olish uchun. Keyin, -\frac{50}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{100}{49}t+\left(-\frac{50}{49}\right)^{2}=-\frac{510204}{49}+\left(-\frac{50}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{50}{49} kvadratini chiqarish.

$$t^{2}-\frac{100}{49}t+\frac{2500}{2401}=-\frac{510204}{49}+\frac{2500}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{510204}{49} ni \frac{2500}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{100}{49}t+\frac{2500}{2401}=-\frac{24997496}{2401}$$

t^{2}-\frac{100}{49}t+\frac{2500}{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.

$$\left(t-\frac{50}{49}\right)^{2}=-\frac{24997496}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{50}{49}\right)^{2}}=\sqrt{-\frac{24997496}{2401}}$$

Qisqartirish.

$$t-\frac{50}{49}=\frac{2\sqrt{6249374}i}{49} t-\frac{50}{49}=-\frac{2\sqrt{6249374}i}{49}$$

\frac{50}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{50+2\sqrt{6249374}i}{49} t=\frac{-2\sqrt{6249374}i+50}{49}$$