7x+5/2x^2-5/2x=1000 eching

$$7x+ \frac{ 5 }{ 2 } { x }^{ 2 } - \frac{ 5 }{ 2 } x=1000$$

x uchun yechish

$$x = \frac{\sqrt{40081} - 9}{10} \approx 19,120239759$$

$$x=\frac{-\sqrt{40081}-9}{10}\approx -20,920239759$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{9}{2}x ni olish uchun 7x va -\frac{5}{2}x ni birlashtirish.

$$\frac{9}{2}x+\frac{5}{2}x^{2}=1000$$

Ikkala tarafdan 1000 ni ayirish.

$$\frac{9}{2}x+\frac{5}{2}x^{2}-1000=0$$

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.

$$\frac{5}{2}x^{2}+\frac{9}{2}x-1000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{5}{2} ni a, \frac{9}{2} ni b va -1000 ni c bilan almashtiring.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\times \frac{5}{2}\left(-1000\right)}}{2\times \frac{5}{2}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{2} kvadratini chiqarish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\times \frac{5}{2}\left(-1000\right)}}{2\times \frac{5}{2}}$$

-4 ni \frac{5}{2} marotabaga ko'paytirish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-10\left(-1000\right)}}{2\times \frac{5}{2}}$$

-10 ni -1000 marotabaga ko'paytirish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+10000}}{2\times \frac{5}{2}}$$

\frac{81}{4} ni 10000 ga qo'shish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{40081}{4}}}{2\times \frac{5}{2}}$$

\frac{40081}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{2\times \frac{5}{2}}$$

2 ni \frac{5}{2} marotabaga ko'paytirish.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5}$$

x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5} tenglamasini yeching, bunda ± musbat. -\frac{9}{2} ni \frac{\sqrt{40081}}{2} ga qo'shish.

$$x=\frac{\sqrt{40081}-9}{2\times 5}$$

\frac{-9+\sqrt{40081}}{2} ni 5 ga bo'lish.

$$x=\frac{\sqrt{40081}-9}{10}$$

x=\frac{-\frac{9}{2}±\frac{\sqrt{40081}}{2}}{5} tenglamasini yeching, bunda ± manfiy. -\frac{9}{2} dan \frac{\sqrt{40081}}{2} ni ayirish.

$$x=\frac{-\sqrt{40081}-9}{2\times 5}$$

\frac{-9-\sqrt{40081}}{2} ni 5 ga bo'lish.

$$x=\frac{-\sqrt{40081}-9}{10}$$

Tenglama yechildi.

$$x=\frac{\sqrt{40081}-9}{10} x=\frac{-\sqrt{40081}-9}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{9}{2}x ni olish uchun 7x va -\frac{5}{2}x ni birlashtirish.

$$\frac{9}{2}x+\frac{5}{2}x^{2}=1000$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$\frac{5}{2}x^{2}+\frac{9}{2}x=1000$$

Tenglamaning ikki tarafini \frac{5}{2} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{5}{2}x^{2}+\frac{9}{2}x}{\frac{5}{2}}=\frac{1000}{\frac{5}{2}}$$

\frac{5}{2} ga bo'lish \frac{5}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{\frac{9}{2}}{\frac{5}{2}}x=\frac{1000}{\frac{5}{2}}$$

\frac{9}{2} ni \frac{5}{2} ga bo'lish \frac{9}{2} ga k'paytirish \frac{5}{2} ga qaytarish.

$$x^{2}+\frac{9}{5}x=\frac{1000}{\frac{5}{2}}$$

1000 ni \frac{5}{2} ga bo'lish 1000 ga k'paytirish \frac{5}{2} ga qaytarish.

$$x^{2}+\frac{9}{5}x=400$$

\frac{9}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{10} olish uchun. Keyin, \frac{9}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{9}{5}x+\left(\frac{9}{10}\right)^{2}=400+\left(\frac{9}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{10} kvadratini chiqarish.

$$x^{2}+\frac{9}{5}x+\frac{81}{100}=400+\frac{81}{100}$$

400 ni \frac{81}{100} ga qo'shish.

$$x^{2}+\frac{9}{5}x+\frac{81}{100}=\frac{40081}{100}$$

x^{2}+\frac{9}{5}x+\frac{81}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{9}{10}\right)^{2}=\frac{40081}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{10}\right)^{2}}=\sqrt{\frac{40081}{100}}$$

Qisqartirish.

$$x+\frac{9}{10}=\frac{\sqrt{40081}}{10} x+\frac{9}{10}=-\frac{\sqrt{40081}}{10}$$

Tenglamaning ikkala tarafidan \frac{9}{10} ni ayirish.

$$x=\frac{\sqrt{40081}-9}{10} x=\frac{-\sqrt{40081}-9}{10}$$