9x^2-14x-14=0 eching

$$9 x ^ { 2 } - 14 x - 14 = 0$$

x uchun yechish

$$x = \frac{5 \sqrt{7} + 7}{9} \approx 2,247639617$$

$$x=\frac{7-5\sqrt{7}}{9}\approx -0,692084062$$

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.

$$9x^{2}-14x-14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -14 ni b va -14 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 9\left(-14\right)}}{2\times 9}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 9\left(-14\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-36\left(-14\right)}}{2\times 9}$$

-36 ni -14 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+504}}{2\times 9}$$

196 ni 504 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{700}}{2\times 9}$$

700 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±10\sqrt{7}}{2\times 9}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±10\sqrt{7}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{14±10\sqrt{7}}{18}$$

x=\frac{14±10\sqrt{7}}{18} tenglamasini yeching, bunda ± musbat. 14 ni 10\sqrt{7} ga qo'shish.

$$x=\frac{10\sqrt{7}+14}{18}$$

14+10\sqrt{7} ni 18 ga bo'lish.

$$x=\frac{5\sqrt{7}+7}{9}$$

x=\frac{14±10\sqrt{7}}{18} tenglamasini yeching, bunda ± manfiy. 14 dan 10\sqrt{7} ni ayirish.

$$x=\frac{14-10\sqrt{7}}{18}$$

14-10\sqrt{7} ni 18 ga bo'lish.

$$x=\frac{7-5\sqrt{7}}{9}$$

Tenglama yechildi.

$$x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}$$

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.

$$9x^{2}-14x-14=0$$

14 ni tenglamaning ikkala tarafiga qo'shish.

$$9x^{2}-14x-14-\left(-14\right)=-\left(-14\right)$$

O‘zidan -14 ayirilsa 0 qoladi.

$$9x^{2}-14x=-\left(-14\right)$$

0 dan -14 ni ayirish.

$$9x^{2}-14x=14$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}-14x}{9}=\frac{14}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{14}{9}x=\frac{14}{9}$$

-\frac{14}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{9} olish uchun. Keyin, -\frac{7}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{14}{9}x+\left(-\frac{7}{9}\right)^{2}=\frac{14}{9}+\left(-\frac{7}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{9} kvadratini chiqarish.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{14}{9}+\frac{49}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{14}{9} ni \frac{49}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{175}{81}$$

x^{2}-\frac{14}{9}x+\frac{49}{81} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{9}\right)^{2}=\frac{175}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{9}\right)^{2}}=\sqrt{\frac{175}{81}}$$

Qisqartirish.

$$x-\frac{7}{9}=\frac{5\sqrt{7}}{9} x-\frac{7}{9}=-\frac{5\sqrt{7}}{9}$$

\frac{7}{9} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}$$