2t^2-7t-7=0 eching

$$2 t ^ { 2 } - 7 t - 7 = 0$$

t uchun yechish

$$t = \frac{\sqrt{105} + 7}{4} \approx 4,311737691$$

$$t=\frac{7-\sqrt{105}}{4}\approx -0,811737691$$

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.

$$2t^{2}-7t-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -7 ni b va -7 ni c bilan almashtiring.

$$t=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

-7 kvadratini chiqarish.

$$t=\frac{-\left(-7\right)±\sqrt{49-4\times 2\left(-7\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$t=\frac{-\left(-7\right)±\sqrt{49-8\left(-7\right)}}{2\times 2}$$

-8 ni -7 marotabaga ko'paytirish.

$$t=\frac{-\left(-7\right)±\sqrt{49+56}}{2\times 2}$$

49 ni 56 ga qo'shish.

$$t=\frac{-\left(-7\right)±\sqrt{105}}{2\times 2}$$

-7 ning teskarisi 7 ga teng.

$$t=\frac{7±\sqrt{105}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$t=\frac{7±\sqrt{105}}{4}$$

t=\frac{7±\sqrt{105}}{4} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{105} ga qo'shish.

$$t=\frac{\sqrt{105}+7}{4}$$

t=\frac{7±\sqrt{105}}{4} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{105} ni ayirish.

$$t=\frac{7-\sqrt{105}}{4}$$

Tenglama yechildi.

$$t=\frac{\sqrt{105}+7}{4} t=\frac{7-\sqrt{105}}{4}$$

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.

$$2t^{2}-7t-7=0$$

7 ni tenglamaning ikkala tarafiga qo'shish.

$$2t^{2}-7t-7-\left(-7\right)=-\left(-7\right)$$

O‘zidan -7 ayirilsa 0 qoladi.

$$2t^{2}-7t=-\left(-7\right)$$

0 dan -7 ni ayirish.

$$2t^{2}-7t=7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2t^{2}-7t}{2}=\frac{7}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$t^{2}-\frac{7}{2}t=\frac{7}{2}$$

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{7}{2}t+\left(-\frac{7}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$t^{2}-\frac{7}{2}t+\frac{49}{16}=\frac{7}{2}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{2} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{7}{2}t+\frac{49}{16}=\frac{105}{16}$$

t^{2}-\frac{7}{2}t+\frac{49}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{7}{4}\right)^{2}=\frac{105}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{7}{4}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Qisqartirish.

$$t-\frac{7}{4}=\frac{\sqrt{105}}{4} t-\frac{7}{4}=-\frac{\sqrt{105}}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{105}+7}{4} t=\frac{7-\sqrt{105}}{4}$$