2x^2-10x+7=0 eching

$$2 x ^ { 2 } - 10 x + 7 = 0$$

x uchun yechish

$$x = \frac{\sqrt{11} + 5}{2} \approx 4,158312395$$

$$x=\frac{5-\sqrt{11}}{2}\approx 0,841687605$$

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.

$$2x^{2}-10x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -10 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 2\times 7}}{2\times 2}$$

-10 kvadratini chiqarish.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 2\times 7}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-8\times 7}}{2\times 2}$$

-8 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-56}}{2\times 2}$$

100 ni -56 ga qo'shish.

$$x=\frac{-\left(-10\right)±\sqrt{44}}{2\times 2}$$

44 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-10\right)±2\sqrt{11}}{2\times 2}$$

-10 ning teskarisi 10 ga teng.

$$x=\frac{10±2\sqrt{11}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{10±2\sqrt{11}}{4}$$

x=\frac{10±2\sqrt{11}}{4} tenglamasini yeching, bunda ± musbat. 10 ni 2\sqrt{11} ga qo'shish.

$$x=\frac{2\sqrt{11}+10}{4}$$

10+2\sqrt{11} ni 4 ga bo'lish.

$$x=\frac{\sqrt{11}+5}{2}$$

x=\frac{10±2\sqrt{11}}{4} tenglamasini yeching, bunda ± manfiy. 10 dan 2\sqrt{11} ni ayirish.

$$x=\frac{10-2\sqrt{11}}{4}$$

10-2\sqrt{11} ni 4 ga bo'lish.

$$x=\frac{5-\sqrt{11}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{11}+5}{2} x=\frac{5-\sqrt{11}}{2}$$

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.

$$2x^{2}-10x+7=0$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$2x^{2}-10x+7-7=-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$2x^{2}-10x=-7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-10x}{2}=-\frac{7}{2}$$

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

$$x^{2}+\left(-\frac{10}{2}\right)x=-\frac{7}{2}$$

-10 ni 2 ga bo'lish.

$$x^{2}-5x=-\frac{7}{2}$$

-5 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{2} olish uchun. Keyin, -\frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{7}{2}+\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=-\frac{7}{2}+\frac{25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{2} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-5x+\frac{25}{4}=\frac{11}{4}$$

x^{2}-5x+\frac{25}{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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{11}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{11}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{11}}{2} x-\frac{5}{2}=-\frac{\sqrt{11}}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{11}+5}{2} x=\frac{5-\sqrt{11}}{2}$$