2x^2-7x+4=0 eching

$$2 x ^ { 2 } - 7 x + 4 = 0$$

x uchun yechish

$$x = \frac{\sqrt{17} + 7}{4} \approx 2,780776406$$

$$x=\frac{7-\sqrt{17}}{4}\approx 0,719223594$$

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.

$$2x^{2}-7x+4=0$$

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

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

-7 kvadratini chiqarish.

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

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\times 4}}{2\times 2}$$

-8 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-32}}{2\times 2}$$

49 ni -32 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{17}}{2\times 2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{17}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{17}}{4}$$

x=\frac{7±\sqrt{17}}{4} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}+7}{4}$$

x=\frac{7±\sqrt{17}}{4} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{17} ni ayirish.

$$x=\frac{7-\sqrt{17}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{17}+7}{4} x=\frac{7-\sqrt{17}}{4}$$

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.

$$2x^{2}-7x+4=0$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$2x^{2}-7x+4-4=-4$$

O‘zidan 4 ayirilsa 0 qoladi.

$$2x^{2}-7x=-4$$

Ikki tarafini 2 ga bo‘ling.

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

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

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

-4 ni 2 ga bo'lish.

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

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-2+\frac{49}{16}$$

-2 ni \frac{49}{16} ga qo'shish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{17}{16}$$

x^{2}-\frac{7}{2}x+\frac{49}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{17}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{17}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{\sqrt{17}}{4} x-\frac{7}{4}=-\frac{\sqrt{17}}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{17}+7}{4} x=\frac{7-\sqrt{17}}{4}$$