1/x+4-1/x+1=11/30 eching

$$\frac { 1 } { x + 4 } - \frac { 1 } { x + 1 } = \frac { 11 } { 30 }$$

x uchun yechish (complex solution)

$$x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}\approx -2,5-2,435532423i$$

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2}\approx -2,5+2,435532423i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -4,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 30\left(x+1\right)\left(x+4\right) ga, x+4,x+1,30 ning eng kichik karralisiga ko‘paytiring.

$$30x+30-\left(30x+120\right)=11\left(x+1\right)\left(x+4\right)$$

30x+120 teskarisini topish uchun har birining teskarisini toping.

$$30x+30-30x-120=11\left(x+1\right)\left(x+4\right)$$

0 ni olish uchun 30x va -30x ni birlashtirish.

$$30-120=11\left(x+1\right)\left(x+4\right)$$

-90 olish uchun 30 dan 120 ni ayirish.

$$-90=11\left(x+1\right)\left(x+4\right)$$

11 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-90=\left(11x+11\right)\left(x+4\right)$$

11x+11 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-90=11x^{2}+55x+44$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$11x^{2}+55x+44=-90$$

90 ni ikki tarafga qo’shing.

$$11x^{2}+55x+44+90=0$$

134 olish uchun 44 va 90'ni qo'shing.

$$11x^{2}+55x+134=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 11 ni a, 55 ni b va 134 ni c bilan almashtiring.

$$x=\frac{-55±\sqrt{55^{2}-4\times 11\times 134}}{2\times 11}$$

55 kvadratini chiqarish.

$$x=\frac{-55±\sqrt{3025-4\times 11\times 134}}{2\times 11}$$

-4 ni 11 marotabaga ko'paytirish.

$$x=\frac{-55±\sqrt{3025-44\times 134}}{2\times 11}$$

-44 ni 134 marotabaga ko'paytirish.

$$x=\frac{-55±\sqrt{3025-5896}}{2\times 11}$$

3025 ni -5896 ga qo'shish.

$$x=\frac{-55±\sqrt{-2871}}{2\times 11}$$

-2871 ning kvadrat ildizini chiqarish.

$$x=\frac{-55±3\sqrt{319}i}{2\times 11}$$

2 ni 11 marotabaga ko'paytirish.

$$x=\frac{-55±3\sqrt{319}i}{22}$$

x=\frac{-55±3\sqrt{319}i}{22} tenglamasini yeching, bunda ± musbat. -55 ni 3i\sqrt{319} ga qo'shish.

$$x=\frac{-55+3\sqrt{319}i}{22}$$

-55+3i\sqrt{319} ni 22 ga bo'lish.

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

x=\frac{-55±3\sqrt{319}i}{22} tenglamasini yeching, bunda ± manfiy. -55 dan 3i\sqrt{319} ni ayirish.

$$x=\frac{-3\sqrt{319}i-55}{22}$$

-55-3i\sqrt{319} ni 22 ga bo'lish.

$$x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2} x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -4,-1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 30\left(x+1\right)\left(x+4\right) ga, x+4,x+1,30 ning eng kichik karralisiga ko‘paytiring.

$$30x+30-\left(30x+120\right)=11\left(x+1\right)\left(x+4\right)$$

30x+120 teskarisini topish uchun har birining teskarisini toping.

$$30x+30-30x-120=11\left(x+1\right)\left(x+4\right)$$

0 ni olish uchun 30x va -30x ni birlashtirish.

$$30-120=11\left(x+1\right)\left(x+4\right)$$

-90 olish uchun 30 dan 120 ni ayirish.

$$-90=11\left(x+1\right)\left(x+4\right)$$

11 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-90=\left(11x+11\right)\left(x+4\right)$$

11x+11 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-90=11x^{2}+55x+44$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$11x^{2}+55x+44=-90$$

Ikkala tarafdan 44 ni ayirish.

$$11x^{2}+55x=-90-44$$

-134 olish uchun -90 dan 44 ni ayirish.

$$11x^{2}+55x=-134$$

Ikki tarafini 11 ga bo‘ling.

$$\frac{11x^{2}+55x}{11}=-\frac{134}{11}$$

11 ga bo'lish 11 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{55}{11}x=-\frac{134}{11}$$

55 ni 11 ga bo'lish.

$$x^{2}+5x=-\frac{134}{11}$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=-\frac{134}{11}+\left(\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{2} kvadratini chiqarish.

$$x^{2}+5x+\frac{25}{4}=-\frac{134}{11}+\frac{25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{134}{11} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+5x+\frac{25}{4}=-\frac{261}{44}$$

x^{2}+5x+\frac{25}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{2}\right)^{2}=-\frac{261}{44}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{-\frac{261}{44}}$$

Qisqartirish.

$$x+\frac{5}{2}=\frac{3\sqrt{319}i}{22} x+\frac{5}{2}=-\frac{3\sqrt{319}i}{22}$$

Tenglamaning ikkala tarafidan \frac{5}{2} ni ayirish.

$$x=\frac{3\sqrt{319}i}{22}-\frac{5}{2} x=-\frac{3\sqrt{319}i}{22}-\frac{5}{2}$$