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

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

x uchun yechish (complex solution)

$$x=\frac{9+\sqrt{39919}i}{10}\approx 0,9+19,979739738i$$

$$x=\frac{-\sqrt{39919}i+9}{10}\approx 0,9-19,979739738i$$

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.

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

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

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

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

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

10 ni -1000 marotabaga ko'paytirish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-10000}}{2\left(-\frac{5}{2}\right)}$$

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

$$x=\frac{-\frac{9}{2}±\sqrt{-\frac{39919}{4}}}{2\left(-\frac{5}{2}\right)}$$

-\frac{39919}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{2\left(-\frac{5}{2}\right)}$$

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

$$x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5}$$

x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} tenglamasini yeching, bunda ± musbat. -\frac{9}{2} ni \frac{i\sqrt{39919}}{2} ga qo'shish.

$$x=\frac{-9+\sqrt{39919}i}{-5\times 2}$$

\frac{-9+i\sqrt{39919}}{2} ni -5 ga bo'lish.

$$x=\frac{-\sqrt{39919}i+9}{10}$$

x=\frac{-\frac{9}{2}±\frac{\sqrt{39919}i}{2}}{-5} tenglamasini yeching, bunda ± manfiy. -\frac{9}{2} dan \frac{i\sqrt{39919}}{2} ni ayirish.

$$x=\frac{-\sqrt{39919}i-9}{-5\times 2}$$

\frac{-9-i\sqrt{39919}}{2} ni -5 ga bo'lish.

$$x=\frac{9+\sqrt{39919}i}{10}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{39919}i+9}{10} x=\frac{9+\sqrt{39919}i}{10}$$

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.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=-\frac{39919}{100}$$

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.

$$\left(x-\frac{9}{10}\right)^{2}=-\frac{39919}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{-\frac{39919}{100}}$$

Qisqartirish.

$$x-\frac{9}{10}=\frac{\sqrt{39919}i}{10} x-\frac{9}{10}=-\frac{\sqrt{39919}i}{10}$$

\frac{9}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9+\sqrt{39919}i}{10} x=\frac{-\sqrt{39919}i+9}{10}$$