3x^2-19x-18=0 eching

$$3 { x }^{ 2 } -19x-18=0$$

x uchun yechish

$$x = \frac{\sqrt{577} + 19}{6} \approx 7,170137383$$

$$x=\frac{19-\sqrt{577}}{6}\approx -0,83680405$$

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.

$$3x^{2}-19x-18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -19 ni b va -18 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 3\left(-18\right)}}{2\times 3}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 3\left(-18\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-12\left(-18\right)}}{2\times 3}$$

-12 ni -18 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+216}}{2\times 3}$$

361 ni 216 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{577}}{2\times 3}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±\sqrt{577}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{19±\sqrt{577}}{6}$$

x=\frac{19±\sqrt{577}}{6} tenglamasini yeching, bunda ± musbat. 19 ni \sqrt{577} ga qo'shish.

$$x=\frac{\sqrt{577}+19}{6}$$

x=\frac{19±\sqrt{577}}{6} tenglamasini yeching, bunda ± manfiy. 19 dan \sqrt{577} ni ayirish.

$$x=\frac{19-\sqrt{577}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{577}+19}{6} x=\frac{19-\sqrt{577}}{6}$$

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.

$$3x^{2}-19x-18=0$$

18 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-19x-18-\left(-18\right)=-\left(-18\right)$$

O‘zidan -18 ayirilsa 0 qoladi.

$$3x^{2}-19x=-\left(-18\right)$$

0 dan -18 ni ayirish.

$$3x^{2}-19x=18$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-19x}{3}=\frac{18}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{19}{3}x=\frac{18}{3}$$

18 ni 3 ga bo'lish.

$$x^{2}-\frac{19}{3}x=6$$

-\frac{19}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{6} olish uchun. Keyin, -\frac{19}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{3}x+\left(-\frac{19}{6}\right)^{2}=6+\left(-\frac{19}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{6} kvadratini chiqarish.

$$x^{2}-\frac{19}{3}x+\frac{361}{36}=6+\frac{361}{36}$$

6 ni \frac{361}{36} ga qo'shish.

$$x^{2}-\frac{19}{3}x+\frac{361}{36}=\frac{577}{36}$$

x^{2}-\frac{19}{3}x+\frac{361}{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.

$$\left(x-\frac{19}{6}\right)^{2}=\frac{577}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{6}\right)^{2}}=\sqrt{\frac{577}{36}}$$

Qisqartirish.

$$x-\frac{19}{6}=\frac{\sqrt{577}}{6} x-\frac{19}{6}=-\frac{\sqrt{577}}{6}$$

\frac{19}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{577}+19}{6} x=\frac{19-\sqrt{577}}{6}$$