37x^2-70x+25=0 eching

$$37 x ^ { 2 } - 70 x + 25 = 0$$

x uchun yechish

$$x = \frac{10 \sqrt{3} + 35}{37} \approx 1,414067786$$

$$x=\frac{35-10\sqrt{3}}{37}\approx 0,477824106$$

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.

$$37x^{2}-70x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 37 ni a, -70 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 37\times 25}}{2\times 37}$$

-70 kvadratini chiqarish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 37\times 25}}{2\times 37}$$

-4 ni 37 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-148\times 25}}{2\times 37}$$

-148 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-3700}}{2\times 37}$$

4900 ni -3700 ga qo'shish.

$$x=\frac{-\left(-70\right)±\sqrt{1200}}{2\times 37}$$

1200 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-70\right)±20\sqrt{3}}{2\times 37}$$

-70 ning teskarisi 70 ga teng.

$$x=\frac{70±20\sqrt{3}}{2\times 37}$$

2 ni 37 marotabaga ko'paytirish.

$$x=\frac{70±20\sqrt{3}}{74}$$

x=\frac{70±20\sqrt{3}}{74} tenglamasini yeching, bunda ± musbat. 70 ni 20\sqrt{3} ga qo'shish.

$$x=\frac{20\sqrt{3}+70}{74}$$

70+20\sqrt{3} ni 74 ga bo'lish.

$$x=\frac{10\sqrt{3}+35}{37}$$

x=\frac{70±20\sqrt{3}}{74} tenglamasini yeching, bunda ± manfiy. 70 dan 20\sqrt{3} ni ayirish.

$$x=\frac{70-20\sqrt{3}}{74}$$

70-20\sqrt{3} ni 74 ga bo'lish.

$$x=\frac{35-10\sqrt{3}}{37}$$

Tenglama yechildi.

$$x=\frac{10\sqrt{3}+35}{37} x=\frac{35-10\sqrt{3}}{37}$$

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.

$$37x^{2}-70x+25=0$$

Tenglamaning ikkala tarafidan 25 ni ayirish.

$$37x^{2}-70x+25-25=-25$$

O‘zidan 25 ayirilsa 0 qoladi.

$$37x^{2}-70x=-25$$

Ikki tarafini 37 ga bo‘ling.

$$\frac{37x^{2}-70x}{37}=-\frac{25}{37}$$

37 ga bo'lish 37 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{70}{37}x=-\frac{25}{37}$$

-\frac{70}{37} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{35}{37} olish uchun. Keyin, -\frac{35}{37} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{70}{37}x+\left(-\frac{35}{37}\right)^{2}=-\frac{25}{37}+\left(-\frac{35}{37}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{35}{37} kvadratini chiqarish.

$$x^{2}-\frac{70}{37}x+\frac{1225}{1369}=-\frac{25}{37}+\frac{1225}{1369}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{37} ni \frac{1225}{1369} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{70}{37}x+\frac{1225}{1369}=\frac{300}{1369}$$

x^{2}-\frac{70}{37}x+\frac{1225}{1369} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{35}{37}\right)^{2}=\frac{300}{1369}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{35}{37}\right)^{2}}=\sqrt{\frac{300}{1369}}$$

Qisqartirish.

$$x-\frac{35}{37}=\frac{10\sqrt{3}}{37} x-\frac{35}{37}=-\frac{10\sqrt{3}}{37}$$

\frac{35}{37} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{10\sqrt{3}+35}{37} x=\frac{35-10\sqrt{3}}{37}$$