3x^2+25x=125 eching

$$3 x ^ { 2 } + 25 x = 125$$

x uchun yechish

$$x = \frac{5 \sqrt{85} - 25}{6} \approx 3,516287048$$

$$x=\frac{-5\sqrt{85}-25}{6}\approx -11,849620381$$

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.

$$3x^{2}+25x=125$$

Tenglamaning ikkala tarafidan 125 ni ayirish.

$$3x^{2}+25x-125=125-125$$

O‘zidan 125 ayirilsa 0 qoladi.

$$3x^{2}+25x-125=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 25 ni b va -125 ni c bilan almashtiring.

$$x=\frac{-25±\sqrt{25^{2}-4\times 3\left(-125\right)}}{2\times 3}$$

25 kvadratini chiqarish.

$$x=\frac{-25±\sqrt{625-4\times 3\left(-125\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-25±\sqrt{625-12\left(-125\right)}}{2\times 3}$$

-12 ni -125 marotabaga ko'paytirish.

$$x=\frac{-25±\sqrt{625+1500}}{2\times 3}$$

625 ni 1500 ga qo'shish.

$$x=\frac{-25±\sqrt{2125}}{2\times 3}$$

2125 ning kvadrat ildizini chiqarish.

$$x=\frac{-25±5\sqrt{85}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-25±5\sqrt{85}}{6}$$

x=\frac{-25±5\sqrt{85}}{6} tenglamasini yeching, bunda ± musbat. -25 ni 5\sqrt{85} ga qo'shish.

$$x=\frac{5\sqrt{85}-25}{6}$$

x=\frac{-25±5\sqrt{85}}{6} tenglamasini yeching, bunda ± manfiy. -25 dan 5\sqrt{85} ni ayirish.

$$x=\frac{-5\sqrt{85}-25}{6}$$

Tenglama yechildi.

$$x=\frac{5\sqrt{85}-25}{6} x=\frac{-5\sqrt{85}-25}{6}$$

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.

$$3x^{2}+25x=125$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+25x}{3}=\frac{125}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{25}{3}x=\frac{125}{3}$$

\frac{25}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{25}{6} olish uchun. Keyin, \frac{25}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{25}{3}x+\left(\frac{25}{6}\right)^{2}=\frac{125}{3}+\left(\frac{25}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{25}{6} kvadratini chiqarish.

$$x^{2}+\frac{25}{3}x+\frac{625}{36}=\frac{125}{3}+\frac{625}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{125}{3} ni \frac{625}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{25}{3}x+\frac{625}{36}=\frac{2125}{36}$$

x^{2}+\frac{25}{3}x+\frac{625}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{25}{6}\right)^{2}=\frac{2125}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{25}{6}\right)^{2}}=\sqrt{\frac{2125}{36}}$$

Qisqartirish.

$$x+\frac{25}{6}=\frac{5\sqrt{85}}{6} x+\frac{25}{6}=-\frac{5\sqrt{85}}{6}$$

Tenglamaning ikkala tarafidan \frac{25}{6} ni ayirish.

$$x=\frac{5\sqrt{85}-25}{6} x=\frac{-5\sqrt{85}-25}{6}$$