x^2+49x=360 eching

$$x ^ { 2 } + 49 x = 360$$

x uchun yechish

$$x = \frac{\sqrt{3841} - 49}{2} \approx 6,487900865$$

$$x=\frac{-\sqrt{3841}-49}{2}\approx -55,487900865$$

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.

$$x^{2}+49x=360$$

Tenglamaning ikkala tarafidan 360 ni ayirish.

$$x^{2}+49x-360=360-360$$

O‘zidan 360 ayirilsa 0 qoladi.

$$x^{2}+49x-360=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 49 ni b va -360 ni c bilan almashtiring.

$$x=\frac{-49±\sqrt{49^{2}-4\left(-360\right)}}{2}$$

49 kvadratini chiqarish.

$$x=\frac{-49±\sqrt{2401-4\left(-360\right)}}{2}$$

-4 ni -360 marotabaga ko'paytirish.

$$x=\frac{-49±\sqrt{2401+1440}}{2}$$

2401 ni 1440 ga qo'shish.

$$x=\frac{-49±\sqrt{3841}}{2}$$

x=\frac{-49±\sqrt{3841}}{2} tenglamasini yeching, bunda ± musbat. -49 ni \sqrt{3841} ga qo'shish.

$$x=\frac{\sqrt{3841}-49}{2}$$

x=\frac{-49±\sqrt{3841}}{2} tenglamasini yeching, bunda ± manfiy. -49 dan \sqrt{3841} ni ayirish.

$$x=\frac{-\sqrt{3841}-49}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{3841}-49}{2} x=\frac{-\sqrt{3841}-49}{2}$$

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.

$$x^{2}+49x=360$$

49 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{49}{2} olish uchun. Keyin, \frac{49}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+49x+\left(\frac{49}{2}\right)^{2}=360+\left(\frac{49}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{49}{2} kvadratini chiqarish.

$$x^{2}+49x+\frac{2401}{4}=360+\frac{2401}{4}$$

360 ni \frac{2401}{4} ga qo'shish.

$$x^{2}+49x+\frac{2401}{4}=\frac{3841}{4}$$

x^{2}+49x+\frac{2401}{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.

$$\left(x+\frac{49}{2}\right)^{2}=\frac{3841}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{49}{2}\right)^{2}}=\sqrt{\frac{3841}{4}}$$

Qisqartirish.

$$x+\frac{49}{2}=\frac{\sqrt{3841}}{2} x+\frac{49}{2}=-\frac{\sqrt{3841}}{2}$$

Tenglamaning ikkala tarafidan \frac{49}{2} ni ayirish.

$$x=\frac{\sqrt{3841}-49}{2} x=\frac{-\sqrt{3841}-49}{2}$$