x^2-7x+3=0 eching

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

x uchun yechish

$$x = \frac{\sqrt{37} + 7}{2} \approx 6,541381265$$

$$x=\frac{7-\sqrt{37}}{2}\approx 0,458618735$$

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -7 ni b va 3 ni c bilan almashtiring.

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

-7 kvadratini chiqarish.

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

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-12}}{2}$$

49 ni -12 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{37}}{2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{37}}{2}$$

x=\frac{7±\sqrt{37}}{2} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{37} ga qo'shish.

$$x=\frac{\sqrt{37}+7}{2}$$

x=\frac{7±\sqrt{37}}{2} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{37} ni ayirish.

$$x=\frac{7-\sqrt{37}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$

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.

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

Tenglamaning ikkala tarafidan 3 ni ayirish.

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

O‘zidan 3 ayirilsa 0 qoladi.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=-3+\frac{49}{4}$$

-3 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{37}{4}$$

x^{2}-7x+\frac{49}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{37}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{\sqrt{37}}{2} x-\frac{7}{2}=-\frac{\sqrt{37}}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{37}+7}{2} x=\frac{7-\sqrt{37}}{2}$$