3x^2+7x+3=0 eching

$$3 x ^ { 2 } + 7 x + 3 = 0$$

x uchun yechish

$$x=\frac{\sqrt{13}-7}{6}\approx -0,565741454$$

$$x=\frac{-\sqrt{13}-7}{6}\approx -1,767591879$$

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.

$$3x^{2}+7x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 7 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 3\times 3}}{2\times 3}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 3\times 3}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-12\times 3}}{2\times 3}$$

-12 ni 3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-36}}{2\times 3}$$

49 ni -36 ga qo'shish.

$$x=\frac{-7±\sqrt{13}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{13}}{6}$$

x=\frac{-7±\sqrt{13}}{6} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{13} ga qo'shish.

$$x=\frac{\sqrt{13}-7}{6}$$

x=\frac{-7±\sqrt{13}}{6} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{13} ni ayirish.

$$x=\frac{-\sqrt{13}-7}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{13}-7}{6} x=\frac{-\sqrt{13}-7}{6}$$

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.

$$3x^{2}+7x+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$3x^{2}+7x+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$3x^{2}+7x=-3$$

Ikki tarafini 3 ga bo‘ling.

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

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

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

-3 ni 3 ga bo'lish.

$$x^{2}+\frac{7}{3}x=-1$$

\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.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=-1+\left(\frac{7}{6}\right)^{2}$$

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

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=-1+\frac{49}{36}$$

-1 ni \frac{49}{36} ga qo'shish.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{13}{36}$$

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.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{13}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{13}{36}}$$

Qisqartirish.

$$x+\frac{7}{6}=\frac{\sqrt{13}}{6} x+\frac{7}{6}=-\frac{\sqrt{13}}{6}$$

Tenglamaning ikkala tarafidan \frac{7}{6} ni ayirish.

$$x=\frac{\sqrt{13}-7}{6} x=\frac{-\sqrt{13}-7}{6}$$