-51t+49t^2=105 eching

$$-51t+49 { t }^{ 2 } = 105$$

t uchun yechish

$$t = \frac{\sqrt{23181} + 51}{98} \approx 2,074011008$$

$$t=\frac{51-\sqrt{23181}}{98}\approx -1,033194681$$

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.

$$49t^{2}-51t=105$$

Tenglamaning ikkala tarafidan 105 ni ayirish.

$$49t^{2}-51t-105=105-105$$

O‘zidan 105 ayirilsa 0 qoladi.

$$49t^{2}-51t-105=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, -51 ni b va -105 ni c bilan almashtiring.

$$t=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\times 49\left(-105\right)}}{2\times 49}$$

-51 kvadratini chiqarish.

$$t=\frac{-\left(-51\right)±\sqrt{2601-4\times 49\left(-105\right)}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$t=\frac{-\left(-51\right)±\sqrt{2601-196\left(-105\right)}}{2\times 49}$$

-196 ni -105 marotabaga ko'paytirish.

$$t=\frac{-\left(-51\right)±\sqrt{2601+20580}}{2\times 49}$$

2601 ni 20580 ga qo'shish.

$$t=\frac{-\left(-51\right)±\sqrt{23181}}{2\times 49}$$

-51 ning teskarisi 51 ga teng.

$$t=\frac{51±\sqrt{23181}}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$t=\frac{51±\sqrt{23181}}{98}$$

t=\frac{51±\sqrt{23181}}{98} tenglamasini yeching, bunda ± musbat. 51 ni \sqrt{23181} ga qo'shish.

$$t=\frac{\sqrt{23181}+51}{98}$$

t=\frac{51±\sqrt{23181}}{98} tenglamasini yeching, bunda ± manfiy. 51 dan \sqrt{23181} ni ayirish.

$$t=\frac{51-\sqrt{23181}}{98}$$

Tenglama yechildi.

$$t=\frac{\sqrt{23181}+51}{98} t=\frac{51-\sqrt{23181}}{98}$$

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.

$$49t^{2}-51t=105$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49t^{2}-51t}{49}=\frac{105}{49}$$

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

$$t^{2}-\frac{51}{49}t=\frac{105}{49}$$

\frac{105}{49} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{51}{49}t=\frac{15}{7}$$

-\frac{51}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{51}{98} olish uchun. Keyin, -\frac{51}{98} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{51}{49}t+\left(-\frac{51}{98}\right)^{2}=\frac{15}{7}+\left(-\frac{51}{98}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{51}{98} kvadratini chiqarish.

$$t^{2}-\frac{51}{49}t+\frac{2601}{9604}=\frac{15}{7}+\frac{2601}{9604}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{7} ni \frac{2601}{9604} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{51}{49}t+\frac{2601}{9604}=\frac{23181}{9604}$$

t^{2}-\frac{51}{49}t+\frac{2601}{9604} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{51}{98}\right)^{2}=\frac{23181}{9604}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{51}{98}\right)^{2}}=\sqrt{\frac{23181}{9604}}$$

Qisqartirish.

$$t-\frac{51}{98}=\frac{\sqrt{23181}}{98} t-\frac{51}{98}=-\frac{\sqrt{23181}}{98}$$

\frac{51}{98} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{23181}+51}{98} t=\frac{51-\sqrt{23181}}{98}$$