25x^2-19x-3=0 eching

$$25 x ^ { 2 } - 19 x - 3 = 0$$

x uchun yechish

$$x=\frac{\sqrt{661}+19}{50}\approx 0,894198405$$

$$x=\frac{19-\sqrt{661}}{50}\approx -0,134198405$$

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.

$$25x^{2}-19x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 25 ni a, -19 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 25\left(-3\right)}}{2\times 25}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 25\left(-3\right)}}{2\times 25}$$

-4 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-100\left(-3\right)}}{2\times 25}$$

-100 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+300}}{2\times 25}$$

361 ni 300 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{661}}{2\times 25}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±\sqrt{661}}{2\times 25}$$

2 ni 25 marotabaga ko'paytirish.

$$x=\frac{19±\sqrt{661}}{50}$$

x=\frac{19±\sqrt{661}}{50} tenglamasini yeching, bunda ± musbat. 19 ni \sqrt{661} ga qo'shish.

$$x=\frac{\sqrt{661}+19}{50}$$

x=\frac{19±\sqrt{661}}{50} tenglamasini yeching, bunda ± manfiy. 19 dan \sqrt{661} ni ayirish.

$$x=\frac{19-\sqrt{661}}{50}$$

Tenglama yechildi.

$$x=\frac{\sqrt{661}+19}{50} x=\frac{19-\sqrt{661}}{50}$$

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.

$$25x^{2}-19x-3=0$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$25x^{2}-19x-3-\left(-3\right)=-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$25x^{2}-19x=-\left(-3\right)$$

0 dan -3 ni ayirish.

$$25x^{2}-19x=3$$

Ikki tarafini 25 ga bo‘ling.

$$\frac{25x^{2}-19x}{25}=\frac{3}{25}$$

25 ga bo'lish 25 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{19}{25}x=\frac{3}{25}$$

-\frac{19}{25} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{50} olish uchun. Keyin, -\frac{19}{50} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{25}x+\left(-\frac{19}{50}\right)^{2}=\frac{3}{25}+\left(-\frac{19}{50}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{50} kvadratini chiqarish.

$$x^{2}-\frac{19}{25}x+\frac{361}{2500}=\frac{3}{25}+\frac{361}{2500}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{25} ni \frac{361}{2500} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{19}{25}x+\frac{361}{2500}=\frac{661}{2500}$$

x^{2}-\frac{19}{25}x+\frac{361}{2500} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{19}{50}\right)^{2}=\frac{661}{2500}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{50}\right)^{2}}=\sqrt{\frac{661}{2500}}$$

Qisqartirish.

$$x-\frac{19}{50}=\frac{\sqrt{661}}{50} x-\frac{19}{50}=-\frac{\sqrt{661}}{50}$$

\frac{19}{50} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{661}+19}{50} x=\frac{19-\sqrt{661}}{50}$$