5x+9/8x-1-5x+1/3x-1=1 eching

$$\frac { 5 x + 9 } { 8 x - 1 } - \frac { 5 x + 1 } { 3 x - 1 } = 1$$

x uchun yechish (complex solution)

$$x=\frac{-6\sqrt{6}i+15}{49}\approx 0,306122449-0,29993752i$$

$$x=\frac{15+6\sqrt{6}i}{49}\approx 0,306122449+0,29993752i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati \frac{1}{8},\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3x-1\right)\left(8x-1\right) ga, 8x-1,3x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

3x-1 ga 5x+9 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

8x-1 ga 5x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

40x^{2}+3x-1 teskarisini topish uchun har birining teskarisini toping.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

-25x^{2} ni olish uchun 15x^{2} va -40x^{2} ni birlashtirish.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

19x ni olish uchun 22x va -3x ni birlashtirish.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

-8 olish uchun -9 va 1'ni qo'shing.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

3x-1 ga 8x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

-49x^{2} ni olish uchun -25x^{2} va -24x^{2} ni birlashtirish.

$$-49x^{2}+19x-8=-11x+1$$

11x ni ikki tarafga qo’shing.

$$-49x^{2}+19x-8+11x=1$$

30x ni olish uchun 19x va 11x ni birlashtirish.

$$-49x^{2}+30x-8=1$$

Ikkala tarafdan 1 ni ayirish.

$$-49x^{2}+30x-8-1=0$$

-9 olish uchun -8 dan 1 ni ayirish.

$$-49x^{2}+30x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 30 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\left(-49\right)\left(-9\right)}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+196\left(-9\right)}}{2\left(-49\right)}$$

196 ni -9 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-1764}}{2\left(-49\right)}$$

900 ni -1764 ga qo'shish.

$$x=\frac{-30±\sqrt{-864}}{2\left(-49\right)}$$

-864 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±12\sqrt{6}i}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{-30±12\sqrt{6}i}{-98}$$

x=\frac{-30±12\sqrt{6}i}{-98} tenglamasini yeching, bunda ± musbat. -30 ni 12i\sqrt{6} ga qo'shish.

$$x=\frac{-30+12\sqrt{6}i}{-98}$$

-30+12i\sqrt{6} ni -98 ga bo'lish.

$$x=\frac{-6\sqrt{6}i+15}{49}$$

x=\frac{-30±12\sqrt{6}i}{-98} tenglamasini yeching, bunda ± manfiy. -30 dan 12i\sqrt{6} ni ayirish.

$$x=\frac{-12\sqrt{6}i-30}{-98}$$

-30-12i\sqrt{6} ni -98 ga bo'lish.

$$x=\frac{15+6\sqrt{6}i}{49}$$

Tenglama yechildi.

$$x=\frac{-6\sqrt{6}i+15}{49} x=\frac{15+6\sqrt{6}i}{49}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati \frac{1}{8},\frac{1}{3} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(3x-1\right)\left(8x-1\right) ga, 8x-1,3x-1 ning eng kichik karralisiga ko‘paytiring.

$$\left(3x-1\right)\left(5x+9\right)-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

3x-1 ga 5x+9 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$15x^{2}+22x-9-\left(8x-1\right)\left(5x+1\right)=\left(3x-1\right)\left(8x-1\right)$$

8x-1 ga 5x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$15x^{2}+22x-9-\left(40x^{2}+3x-1\right)=\left(3x-1\right)\left(8x-1\right)$$

40x^{2}+3x-1 teskarisini topish uchun har birining teskarisini toping.

$$15x^{2}+22x-9-40x^{2}-3x+1=\left(3x-1\right)\left(8x-1\right)$$

-25x^{2} ni olish uchun 15x^{2} va -40x^{2} ni birlashtirish.

$$-25x^{2}+22x-9-3x+1=\left(3x-1\right)\left(8x-1\right)$$

19x ni olish uchun 22x va -3x ni birlashtirish.

$$-25x^{2}+19x-9+1=\left(3x-1\right)\left(8x-1\right)$$

-8 olish uchun -9 va 1'ni qo'shing.

$$-25x^{2}+19x-8=\left(3x-1\right)\left(8x-1\right)$$

3x-1 ga 8x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-25x^{2}+19x-8=24x^{2}-11x+1$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$-25x^{2}+19x-8-24x^{2}=-11x+1$$

-49x^{2} ni olish uchun -25x^{2} va -24x^{2} ni birlashtirish.

$$-49x^{2}+19x-8=-11x+1$$

11x ni ikki tarafga qo’shing.

$$-49x^{2}+19x-8+11x=1$$

30x ni olish uchun 19x va 11x ni birlashtirish.

$$-49x^{2}+30x-8=1$$

8 ni ikki tarafga qo’shing.

$$-49x^{2}+30x=1+8$$

9 olish uchun 1 va 8'ni qo'shing.

$$-49x^{2}+30x=9$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}+30x}{-49}=\frac{9}{-49}$$

-49 ga bo'lish -49 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{30}{-49}x=\frac{9}{-49}$$

30 ni -49 ga bo'lish.

$$x^{2}-\frac{30}{49}x=\frac{9}{-49}$$

9 ni -49 ga bo'lish.

$$x^{2}-\frac{30}{49}x=-\frac{9}{49}$$

-\frac{30}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{49} olish uchun. Keyin, -\frac{15}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{30}{49}x+\left(-\frac{15}{49}\right)^{2}=-\frac{9}{49}+\left(-\frac{15}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{49} kvadratini chiqarish.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{9}{49}+\frac{225}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{49} ni \frac{225}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{30}{49}x+\frac{225}{2401}=-\frac{216}{2401}$$

x^{2}-\frac{30}{49}x+\frac{225}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{49}\right)^{2}=-\frac{216}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{49}\right)^{2}}=\sqrt{-\frac{216}{2401}}$$

Qisqartirish.

$$x-\frac{15}{49}=\frac{6\sqrt{6}i}{49} x-\frac{15}{49}=-\frac{6\sqrt{6}i}{49}$$

\frac{15}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15+6\sqrt{6}i}{49} x=\frac{-6\sqrt{6}i+15}{49}$$