6x^2-14x-9=0 eching

$$6 x ^ { 2 } - 14 x - 9 = 0$$

x uchun yechish

$$x = \frac{\sqrt{103} + 7}{6} \approx 2,858148594$$

$$x=\frac{7-\sqrt{103}}{6}\approx -0,524815261$$

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.

$$6x^{2}-14x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -14 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 6\left(-9\right)}}{2\times 6}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 6\left(-9\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-24\left(-9\right)}}{2\times 6}$$

-24 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+216}}{2\times 6}$$

196 ni 216 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{412}}{2\times 6}$$

412 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2\sqrt{103}}{2\times 6}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2\sqrt{103}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{14±2\sqrt{103}}{12}$$

x=\frac{14±2\sqrt{103}}{12} tenglamasini yeching, bunda ± musbat. 14 ni 2\sqrt{103} ga qo'shish.

$$x=\frac{2\sqrt{103}+14}{12}$$

14+2\sqrt{103} ni 12 ga bo'lish.

$$x=\frac{\sqrt{103}+7}{6}$$

x=\frac{14±2\sqrt{103}}{12} tenglamasini yeching, bunda ± manfiy. 14 dan 2\sqrt{103} ni ayirish.

$$x=\frac{14-2\sqrt{103}}{12}$$

14-2\sqrt{103} ni 12 ga bo'lish.

$$x=\frac{7-\sqrt{103}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{103}+7}{6} x=\frac{7-\sqrt{103}}{6}$$

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.

$$6x^{2}-14x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$6x^{2}-14x-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$6x^{2}-14x=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$6x^{2}-14x=9$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-14x}{6}=\frac{9}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{14}{6}\right)x=\frac{9}{6}$$

\frac{-14}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{3}x=\frac{9}{6}$$

\frac{9}{6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{3}x=\frac{3}{2}$$

-\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{6} olish uchun. Keyin, -\frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=\frac{3}{2}+\left(-\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{6} kvadratini chiqarish.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{3}{2}+\frac{49}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{2} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{103}{36}$$

x^{2}-\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{103}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{103}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{\sqrt{103}}{6} x-\frac{7}{6}=-\frac{\sqrt{103}}{6}$$

\frac{7}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{103}+7}{6} x=\frac{7-\sqrt{103}}{6}$$