-49t^2+98t+100=0 eching

$$- 49 t ^ { 2 } + 98 t + 100 = 0$$

t uchun yechish

$$t=\frac{\sqrt{149}}{7}+1\approx 2,743793659$$

$$t=-\frac{\sqrt{149}}{7}+1\approx -0,743793659$$

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.

$$-49t^{2}+98t+100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 98 ni b va 100 ni c bilan almashtiring.

$$t=\frac{-98±\sqrt{98^{2}-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

98 kvadratini chiqarish.

$$t=\frac{-98±\sqrt{9604-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$t=\frac{-98±\sqrt{9604+196\times 100}}{2\left(-49\right)}$$

196 ni 100 marotabaga ko'paytirish.

$$t=\frac{-98±\sqrt{9604+19600}}{2\left(-49\right)}$$

9604 ni 19600 ga qo'shish.

$$t=\frac{-98±\sqrt{29204}}{2\left(-49\right)}$$

29204 ning kvadrat ildizini chiqarish.

$$t=\frac{-98±14\sqrt{149}}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$t=\frac{-98±14\sqrt{149}}{-98}$$

t=\frac{-98±14\sqrt{149}}{-98} tenglamasini yeching, bunda ± musbat. -98 ni 14\sqrt{149} ga qo'shish.

$$t=\frac{14\sqrt{149}-98}{-98}$$

-98+14\sqrt{149} ni -98 ga bo'lish.

$$t=-\frac{\sqrt{149}}{7}+1$$

t=\frac{-98±14\sqrt{149}}{-98} tenglamasini yeching, bunda ± manfiy. -98 dan 14\sqrt{149} ni ayirish.

$$t=\frac{-14\sqrt{149}-98}{-98}$$

-98-14\sqrt{149} ni -98 ga bo'lish.

$$t=\frac{\sqrt{149}}{7}+1$$

Tenglama yechildi.

$$t=-\frac{\sqrt{149}}{7}+1 t=\frac{\sqrt{149}}{7}+1$$

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.

$$-49t^{2}+98t+100=0$$

Tenglamaning ikkala tarafidan 100 ni ayirish.

$$-49t^{2}+98t+100-100=-100$$

O‘zidan 100 ayirilsa 0 qoladi.

$$-49t^{2}+98t=-100$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49t^{2}+98t}{-49}=-\frac{100}{-49}$$

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

$$t^{2}+\frac{98}{-49}t=-\frac{100}{-49}$$

98 ni -49 ga bo'lish.

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

-100 ni -49 ga bo'lish.

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

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-2t+1=\frac{100}{49}+1$$

\frac{100}{49} ni 1 ga qo'shish.

$$t^{2}-2t+1=\frac{149}{49}$$

t^{2}-2t+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-1\right)^{2}=\frac{149}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-1\right)^{2}}=\sqrt{\frac{149}{49}}$$

Qisqartirish.

$$t-1=\frac{\sqrt{149}}{7} t-1=-\frac{\sqrt{149}}{7}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{149}}{7}+1 t=-\frac{\sqrt{149}}{7}+1$$