20=-49t^2+20t+130 eching

$$20 = - 49 t ^ { 2 } + 20 t + 130$$

t uchun yechish

$$t = \frac{3 \sqrt{610} + 10}{49} \approx 1,716214984$$

$$t=\frac{10-3\sqrt{610}}{49}\approx -1,308051719$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-49t^{2}+20t+130=20$$

Ikkala tarafdan 20 ni ayirish.

$$-49t^{2}+20t+130-20=0$$

110 olish uchun 130 dan 20 ni ayirish.

$$-49t^{2}+20t+110=0$$

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 110 ni c bilan almashtiring.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

20 kvadratini chiqarish.

$$t=\frac{-20±\sqrt{400-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$t=\frac{-20±\sqrt{400+196\times 110}}{2\left(-49\right)}$$

196 ni 110 marotabaga ko'paytirish.

$$t=\frac{-20±\sqrt{400+21560}}{2\left(-49\right)}$$

400 ni 21560 ga qo'shish.

$$t=\frac{-20±\sqrt{21960}}{2\left(-49\right)}$$

21960 ning kvadrat ildizini chiqarish.

$$t=\frac{-20±6\sqrt{610}}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$t=\frac{-20±6\sqrt{610}}{-98}$$

t=\frac{-20±6\sqrt{610}}{-98} tenglamasini yeching, bunda ± musbat. -20 ni 6\sqrt{610} ga qo'shish.

$$t=\frac{6\sqrt{610}-20}{-98}$$

-20+6\sqrt{610} ni -98 ga bo'lish.

$$t=\frac{10-3\sqrt{610}}{49}$$

t=\frac{-20±6\sqrt{610}}{-98} tenglamasini yeching, bunda ± manfiy. -20 dan 6\sqrt{610} ni ayirish.

$$t=\frac{-6\sqrt{610}-20}{-98}$$

-20-6\sqrt{610} ni -98 ga bo'lish.

$$t=\frac{3\sqrt{610}+10}{49}$$

Tenglama yechildi.

$$t=\frac{10-3\sqrt{610}}{49} t=\frac{3\sqrt{610}+10}{49}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-49t^{2}+20t+130=20$$

Ikkala tarafdan 130 ni ayirish.

$$-49t^{2}+20t=20-130$$

-110 olish uchun 20 dan 130 ni ayirish.

$$-49t^{2}+20t=-110$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49t^{2}+20t}{-49}=-\frac{110}{-49}$$

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

$$t^{2}+\frac{20}{-49}t=-\frac{110}{-49}$$

20 ni -49 ga bo'lish.

$$t^{2}-\frac{20}{49}t=-\frac{110}{-49}$$

-110 ni -49 ga bo'lish.

$$t^{2}-\frac{20}{49}t=\frac{110}{49}$$

-\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.

$$t^{2}-\frac{20}{49}t+\left(-\frac{10}{49}\right)^{2}=\frac{110}{49}+\left(-\frac{10}{49}\right)^{2}$$

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

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{110}{49}+\frac{100}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{110}{49} ni \frac{100}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{5490}{2401}$$

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.

$$\left(t-\frac{10}{49}\right)^{2}=\frac{5490}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{10}{49}\right)^{2}}=\sqrt{\frac{5490}{2401}}$$

Qisqartirish.

$$t-\frac{10}{49}=\frac{3\sqrt{610}}{49} t-\frac{10}{49}=-\frac{3\sqrt{610}}{49}$$

\frac{10}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{3\sqrt{610}+10}{49} t=\frac{10-3\sqrt{610}}{49}$$