1/frac{1{x+10}-1/x}=720 eching

$$\frac{ 1 }{ \frac{ 1 }{ x+10 } - \frac{ 1 }{ x } } =720$$

x uchun yechish (complex solution)

$$x=-5+5\sqrt{287}i\approx -5+84,70537173i$$

$$x=-5\sqrt{287}i-5\approx -5-84,70537173i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x+10 va x ning eng kichik umumiy karralisi x\left(x+10\right). \frac{1}{x+10} ni \frac{x}{x} marotabaga ko'paytirish. \frac{1}{x} ni \frac{x+10}{x+10} marotabaga ko'paytirish.

$$\frac{1}{\frac{x}{x\left(x+10\right)}-\frac{x+10}{x\left(x+10\right)}}=720$$

\frac{x}{x\left(x+10\right)} va \frac{x+10}{x\left(x+10\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{\frac{x-\left(x+10\right)}{x\left(x+10\right)}}=720$$

x-\left(x+10\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{1}{\frac{x-x-10}{x\left(x+10\right)}}=720$$

x-x-10 kabi iboralarga o‘xshab birlashtiring.

$$\frac{1}{\frac{-10}{x\left(x+10\right)}}=720$$

x qiymati -10,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{-10}{x\left(x+10\right)} ga bo'lish 1 ga k'paytirish \frac{-10}{x\left(x+10\right)} ga qaytarish.

$$\frac{x\left(x+10\right)}{-10}=720$$

x ga x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{x^{2}+10x}{-10}=720$$

-\frac{1}{10}x^{2}-x natijani olish uchun x^{2}+10x ning har bir ifodasini -10 ga bo‘ling.

$$-\frac{1}{10}x^{2}-x=720$$

Ikkala tarafdan 720 ni ayirish.

$$-\frac{1}{10}x^{2}-x-720=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{1}{10} ni a, -1 ni b va -720 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{1}{10}\right)\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

-4 ni -\frac{1}{10} marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{2}{5}\left(-720\right)}}{2\left(-\frac{1}{10}\right)}$$

\frac{2}{5} ni -720 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-288}}{2\left(-\frac{1}{10}\right)}$$

1 ni -288 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-287}}{2\left(-\frac{1}{10}\right)}$$

-287 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{287}i}{2\left(-\frac{1}{10}\right)}$$

2 ni -\frac{1}{10} marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{287}i}{-\frac{1}{5}}$$

x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{287} ga qo'shish.

$$x=\frac{1+\sqrt{287}i}{-\frac{1}{5}}$$

1+i\sqrt{287} ni -\frac{1}{5} ga bo'lish 1+i\sqrt{287} ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x=-5\sqrt{287}i-5$$

x=\frac{1±\sqrt{287}i}{-\frac{1}{5}} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{287} ni ayirish.

$$x=\frac{-\sqrt{287}i+1}{-\frac{1}{5}}$$

1-i\sqrt{287} ni -\frac{1}{5} ga bo'lish 1-i\sqrt{287} ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x=-5+5\sqrt{287}i$$

Tenglama yechildi.

$$x=-5\sqrt{287}i-5 x=-5+5\sqrt{287}i$$

Kvadratni yakunlash bo'yicha qadamlar

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x+10 va x ning eng kichik umumiy karralisi x\left(x+10\right). \frac{1}{x+10} ni \frac{x}{x} marotabaga ko'paytirish. \frac{1}{x} ni \frac{x+10}{x+10} marotabaga ko'paytirish.

$$\frac{1}{\frac{x}{x\left(x+10\right)}-\frac{x+10}{x\left(x+10\right)}}=720$$

\frac{x}{x\left(x+10\right)} va \frac{x+10}{x\left(x+10\right)} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{\frac{x-\left(x+10\right)}{x\left(x+10\right)}}=720$$

x-\left(x+10\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{1}{\frac{x-x-10}{x\left(x+10\right)}}=720$$

x-x-10 kabi iboralarga o‘xshab birlashtiring.

$$\frac{1}{\frac{-10}{x\left(x+10\right)}}=720$$

x qiymati -10,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. 1 ni \frac{-10}{x\left(x+10\right)} ga bo'lish 1 ga k'paytirish \frac{-10}{x\left(x+10\right)} ga qaytarish.

$$\frac{x\left(x+10\right)}{-10}=720$$

x ga x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{x^{2}+10x}{-10}=720$$

-\frac{1}{10}x^{2}-x natijani olish uchun x^{2}+10x ning har bir ifodasini -10 ga bo‘ling.

$$-\frac{1}{10}x^{2}-x=720$$

Ikkala tarafini -10 ga ko‘paytiring.

$$\frac{-\frac{1}{10}x^{2}-x}{-\frac{1}{10}}=\frac{720}{-\frac{1}{10}}$$

-\frac{1}{10} ga bo'lish -\frac{1}{10} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{1}{-\frac{1}{10}}\right)x=\frac{720}{-\frac{1}{10}}$$

-1 ni -\frac{1}{10} ga bo'lish -1 ga k'paytirish -\frac{1}{10} ga qaytarish.

$$x^{2}+10x=\frac{720}{-\frac{1}{10}}$$

720 ni -\frac{1}{10} ga bo'lish 720 ga k'paytirish -\frac{1}{10} ga qaytarish.

$$x^{2}+10x=-7200$$

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

$$x^{2}+10x+5^{2}=-7200+5^{2}$$

5 kvadratini chiqarish.

$$x^{2}+10x+25=-7200+25$$

-7200 ni 25 ga qo'shish.

$$x^{2}+10x+25=-7175$$

x^{2}+10x+25 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+5\right)^{2}=-7175$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-7175}$$

Qisqartirish.

$$x+5=5\sqrt{287}i x+5=-5\sqrt{287}i$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$x=-5+5\sqrt{287}i x=-5\sqrt{287}i-5$$