3x^2+45x-354=0 eching

$$3 { x }^{ 2 } +45x-354=0$$

x uchun yechish

$$x = \frac{\sqrt{697} - 15}{2} \approx 5,700378782$$

$$x=\frac{-\sqrt{697}-15}{2}\approx -20,700378782$$

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.

$$3x^{2}+45x-354=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 45 ni b va -354 ni c bilan almashtiring.

$$x=\frac{-45±\sqrt{45^{2}-4\times 3\left(-354\right)}}{2\times 3}$$

45 kvadratini chiqarish.

$$x=\frac{-45±\sqrt{2025-4\times 3\left(-354\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-45±\sqrt{2025-12\left(-354\right)}}{2\times 3}$$

-12 ni -354 marotabaga ko'paytirish.

$$x=\frac{-45±\sqrt{2025+4248}}{2\times 3}$$

2025 ni 4248 ga qo'shish.

$$x=\frac{-45±\sqrt{6273}}{2\times 3}$$

6273 ning kvadrat ildizini chiqarish.

$$x=\frac{-45±3\sqrt{697}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-45±3\sqrt{697}}{6}$$

x=\frac{-45±3\sqrt{697}}{6} tenglamasini yeching, bunda ± musbat. -45 ni 3\sqrt{697} ga qo'shish.

$$x=\frac{3\sqrt{697}-45}{6}$$

-45+3\sqrt{697} ni 6 ga bo'lish.

$$x=\frac{\sqrt{697}-15}{2}$$

x=\frac{-45±3\sqrt{697}}{6} tenglamasini yeching, bunda ± manfiy. -45 dan 3\sqrt{697} ni ayirish.

$$x=\frac{-3\sqrt{697}-45}{6}$$

-45-3\sqrt{697} ni 6 ga bo'lish.

$$x=\frac{-\sqrt{697}-15}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{697}-15}{2} x=\frac{-\sqrt{697}-15}{2}$$

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.

$$3x^{2}+45x-354=0$$

354 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}+45x-354-\left(-354\right)=-\left(-354\right)$$

O‘zidan -354 ayirilsa 0 qoladi.

$$3x^{2}+45x=-\left(-354\right)$$

0 dan -354 ni ayirish.

$$3x^{2}+45x=354$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+45x}{3}=\frac{354}{3}$$

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

$$x^{2}+\frac{45}{3}x=\frac{354}{3}$$

45 ni 3 ga bo'lish.

$$x^{2}+15x=\frac{354}{3}$$

354 ni 3 ga bo'lish.

$$x^{2}+15x=118$$

15 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{15}{2} olish uchun. Keyin, \frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=118+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$x^{2}+15x+\frac{225}{4}=118+\frac{225}{4}$$

118 ni \frac{225}{4} ga qo'shish.

$$x^{2}+15x+\frac{225}{4}=\frac{697}{4}$$

x^{2}+15x+\frac{225}{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.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{697}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{697}{4}}$$

Qisqartirish.

$$x+\frac{15}{2}=\frac{\sqrt{697}}{2} x+\frac{15}{2}=-\frac{\sqrt{697}}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$x=\frac{\sqrt{697}-15}{2} x=\frac{-\sqrt{697}-15}{2}$$