(6/25+x)^2x=32 eching

$$( \frac { 6 } { 25 + x } ) ^ { 2 } x = 32$$

x uchun yechish (complex solution)

$$x=\frac{-3\sqrt{791}i-391}{16}\approx -24,4375-5,273385416i$$

$$x=\frac{-391+3\sqrt{791}i}{16}\approx -24,4375+5,273385416i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{6}{25+x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{6^{2}}{\left(25+x\right)^{2}}x=32$$

\frac{6^{2}}{\left(25+x\right)^{2}}x ni yagona kasrga aylantiring.

$$\frac{6^{2}x}{\left(25+x\right)^{2}}=32$$

2 daraja ko‘rsatkichini 6 ga hisoblang va 36 ni qiymatni oling.

$$\frac{36x}{\left(25+x\right)^{2}}=32$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(25+x\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{36x}{625+50x+x^{2}}=32$$

Ikkala tarafdan 32 ni ayirish.

$$\frac{36x}{625+50x+x^{2}}-32=0$$

Faktor: 625+50x+x^{2}.

$$\frac{36x}{\left(x+25\right)^{2}}-32=0$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 32 ni \frac{\left(x+25\right)^{2}}{\left(x+25\right)^{2}} marotabaga ko'paytirish.

$$\frac{36x}{\left(x+25\right)^{2}}-\frac{32\left(x+25\right)^{2}}{\left(x+25\right)^{2}}=0$$

\frac{36x}{\left(x+25\right)^{2}} va \frac{32\left(x+25\right)^{2}}{\left(x+25\right)^{2}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{36x-32\left(x+25\right)^{2}}{\left(x+25\right)^{2}}=0$$

36x-32\left(x+25\right)^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{36x-32x^{2}-1600x-20000}{\left(x+25\right)^{2}}=0$$

36x-32x^{2}-1600x-20000 kabi iboralarga o‘xshab birlashtiring.

$$\frac{-1564x-32x^{2}-20000}{\left(x+25\right)^{2}}=0$$

x qiymati -25 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+25\right)^{2} ga ko'paytirish.

$$-1564x-32x^{2}-20000=0$$

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.

$$-32x^{2}-1564x-20000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -32 ni a, -1564 ni b va -20000 ni c bilan almashtiring.

$$x=\frac{-\left(-1564\right)±\sqrt{\left(-1564\right)^{2}-4\left(-32\right)\left(-20000\right)}}{2\left(-32\right)}$$

-1564 kvadratini chiqarish.

$$x=\frac{-\left(-1564\right)±\sqrt{2446096-4\left(-32\right)\left(-20000\right)}}{2\left(-32\right)}$$

-4 ni -32 marotabaga ko'paytirish.

$$x=\frac{-\left(-1564\right)±\sqrt{2446096+128\left(-20000\right)}}{2\left(-32\right)}$$

128 ni -20000 marotabaga ko'paytirish.

$$x=\frac{-\left(-1564\right)±\sqrt{2446096-2560000}}{2\left(-32\right)}$$

2446096 ni -2560000 ga qo'shish.

$$x=\frac{-\left(-1564\right)±\sqrt{-113904}}{2\left(-32\right)}$$

-113904 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1564\right)±12\sqrt{791}i}{2\left(-32\right)}$$

-1564 ning teskarisi 1564 ga teng.

$$x=\frac{1564±12\sqrt{791}i}{2\left(-32\right)}$$

2 ni -32 marotabaga ko'paytirish.

$$x=\frac{1564±12\sqrt{791}i}{-64}$$

x=\frac{1564±12\sqrt{791}i}{-64} tenglamasini yeching, bunda ± musbat. 1564 ni 12i\sqrt{791} ga qo'shish.

$$x=\frac{1564+12\sqrt{791}i}{-64}$$

1564+12i\sqrt{791} ni -64 ga bo'lish.

$$x=\frac{-3\sqrt{791}i-391}{16}$$

x=\frac{1564±12\sqrt{791}i}{-64} tenglamasini yeching, bunda ± manfiy. 1564 dan 12i\sqrt{791} ni ayirish.

$$x=\frac{-12\sqrt{791}i+1564}{-64}$$

1564-12i\sqrt{791} ni -64 ga bo'lish.

$$x=\frac{-391+3\sqrt{791}i}{16}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{791}i-391}{16} x=\frac{-391+3\sqrt{791}i}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{6}{25+x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{6^{2}}{\left(25+x\right)^{2}}x=32$$

\frac{6^{2}}{\left(25+x\right)^{2}}x ni yagona kasrga aylantiring.

$$\frac{6^{2}x}{\left(25+x\right)^{2}}=32$$

2 daraja ko‘rsatkichini 6 ga hisoblang va 36 ni qiymatni oling.

$$\frac{36x}{\left(25+x\right)^{2}}=32$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(25+x\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{36x}{625+50x+x^{2}}=32$$

x qiymati -25 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x+25\right)^{2} ga ko'paytirish.

$$36x=32\left(x+25\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+25\right)^{2} kengaytirilishi uchun ishlating.

$$36x=32\left(x^{2}+50x+625\right)$$

32 ga x^{2}+50x+625 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x=32x^{2}+1600x+20000$$

Ikkala tarafdan 32x^{2} ni ayirish.

$$36x-32x^{2}=1600x+20000$$

Ikkala tarafdan 1600x ni ayirish.

$$36x-32x^{2}-1600x=20000$$

-1564x ni olish uchun 36x va -1600x ni birlashtirish.

$$-1564x-32x^{2}=20000$$

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

$$-32x^{2}-1564x=20000$$

Ikki tarafini -32 ga bo‘ling.

$$\frac{-32x^{2}-1564x}{-32}=\frac{20000}{-32}$$

-32 ga bo'lish -32 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{1564}{-32}\right)x=\frac{20000}{-32}$$

\frac{-1564}{-32} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{391}{8}x=\frac{20000}{-32}$$

20000 ni -32 ga bo'lish.

$$x^{2}+\frac{391}{8}x=-625$$

\frac{391}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{391}{16} olish uchun. Keyin, \frac{391}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{391}{8}x+\left(\frac{391}{16}\right)^{2}=-625+\left(\frac{391}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{391}{16} kvadratini chiqarish.

$$x^{2}+\frac{391}{8}x+\frac{152881}{256}=-625+\frac{152881}{256}$$

-625 ni \frac{152881}{256} ga qo'shish.

$$x^{2}+\frac{391}{8}x+\frac{152881}{256}=-\frac{7119}{256}$$

x^{2}+\frac{391}{8}x+\frac{152881}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{391}{16}\right)^{2}=-\frac{7119}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{391}{16}\right)^{2}}=\sqrt{-\frac{7119}{256}}$$

Qisqartirish.

$$x+\frac{391}{16}=\frac{3\sqrt{791}i}{16} x+\frac{391}{16}=-\frac{3\sqrt{791}i}{16}$$

Tenglamaning ikkala tarafidan \frac{391}{16} ni ayirish.

$$x=\frac{-391+3\sqrt{791}i}{16} x=\frac{-3\sqrt{791}i-391}{16}$$