49x^2-200x-500=0 eching

$$49 { x }^{ 2 } -200x-500=0$$

x uchun yechish

$$x = \frac{10 \sqrt{345} + 100}{49} \approx 5,831464412$$

$$x=\frac{100-10\sqrt{345}}{49}\approx -1,749831759$$

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.

$$49x^{2}-200x-500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, -200 ni b va -500 ni c bilan almashtiring.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 49\left(-500\right)}}{2\times 49}$$

-200 kvadratini chiqarish.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\times 49\left(-500\right)}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-\left(-200\right)±\sqrt{40000-196\left(-500\right)}}{2\times 49}$$

-196 ni -500 marotabaga ko'paytirish.

$$x=\frac{-\left(-200\right)±\sqrt{40000+98000}}{2\times 49}$$

40000 ni 98000 ga qo'shish.

$$x=\frac{-\left(-200\right)±\sqrt{138000}}{2\times 49}$$

138000 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-200\right)±20\sqrt{345}}{2\times 49}$$

-200 ning teskarisi 200 ga teng.

$$x=\frac{200±20\sqrt{345}}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{200±20\sqrt{345}}{98}$$

x=\frac{200±20\sqrt{345}}{98} tenglamasini yeching, bunda ± musbat. 200 ni 20\sqrt{345} ga qo'shish.

$$x=\frac{20\sqrt{345}+200}{98}$$

200+20\sqrt{345} ni 98 ga bo'lish.

$$x=\frac{10\sqrt{345}+100}{49}$$

x=\frac{200±20\sqrt{345}}{98} tenglamasini yeching, bunda ± manfiy. 200 dan 20\sqrt{345} ni ayirish.

$$x=\frac{200-20\sqrt{345}}{98}$$

200-20\sqrt{345} ni 98 ga bo'lish.

$$x=\frac{100-10\sqrt{345}}{49}$$

Tenglama yechildi.

$$x=\frac{10\sqrt{345}+100}{49} x=\frac{100-10\sqrt{345}}{49}$$

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.

$$49x^{2}-200x-500=0$$

500 ni tenglamaning ikkala tarafiga qo'shish.

$$49x^{2}-200x-500-\left(-500\right)=-\left(-500\right)$$

O‘zidan -500 ayirilsa 0 qoladi.

$$49x^{2}-200x=-\left(-500\right)$$

0 dan -500 ni ayirish.

$$49x^{2}-200x=500$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49x^{2}-200x}{49}=\frac{500}{49}$$

49 ga bo'lish 49 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{200}{49}x=\frac{500}{49}$$

-\frac{200}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{100}{49} olish uchun. Keyin, -\frac{100}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{200}{49}x+\left(-\frac{100}{49}\right)^{2}=\frac{500}{49}+\left(-\frac{100}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{100}{49} kvadratini chiqarish.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=\frac{500}{49}+\frac{10000}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{500}{49} ni \frac{10000}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{200}{49}x+\frac{10000}{2401}=\frac{34500}{2401}$$

x^{2}-\frac{200}{49}x+\frac{10000}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{100}{49}\right)^{2}=\frac{34500}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{100}{49}\right)^{2}}=\sqrt{\frac{34500}{2401}}$$

Qisqartirish.

$$x-\frac{100}{49}=\frac{10\sqrt{345}}{49} x-\frac{100}{49}=-\frac{10\sqrt{345}}{49}$$

\frac{100}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{10\sqrt{345}+100}{49} x=\frac{100-10\sqrt{345}}{49}$$