4x-3/2x+1-102x-1/4x-3=3 eching

$$\frac{ 4x-3 }{ 2x+1 } -10 \frac{ 2x-1 }{ 4x-3 } =3$$

x uchun yechish

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16}\approx 0,598941087$$

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}\approx -0,973941087$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{1}{2},\frac{3}{4} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(4x-3\right)\left(2x+1\right) ga, 2x+1,4x-3 ning eng kichik karralisiga ko‘paytiring.

$$\left(4x-3\right)\left(4x-3\right)-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

\left(4x-3\right)^{2} hosil qilish uchun 4x-3 va 4x-3 ni ko'paytirish.

$$\left(4x-3\right)^{2}-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-3\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

3 ga 4x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=\left(12x-9\right)\left(2x+1\right)$$

12x-9 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=24x^{2}-6x-9$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}=-6x-9$$

6x ni ikki tarafga qo’shing.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x=-9$$

9 ni ikki tarafga qo’shing.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x+9=0$$

-10 ga 2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-24x+9+\left(-20x-10\right)\left(2x-1\right)-24x^{2}+6x+9=0$$

-20x-10 ga 2x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$16x^{2}-24x+9-40x^{2}+10-24x^{2}+6x+9=0$$

-24x^{2} ni olish uchun 16x^{2} va -40x^{2} ni birlashtirish.

$$-24x^{2}-24x+9+10-24x^{2}+6x+9=0$$

19 olish uchun 9 va 10'ni qo'shing.

$$-24x^{2}-24x+19-24x^{2}+6x+9=0$$

-48x^{2} ni olish uchun -24x^{2} va -24x^{2} ni birlashtirish.

$$-48x^{2}-24x+19+6x+9=0$$

-18x ni olish uchun -24x va 6x ni birlashtirish.

$$-48x^{2}-18x+19+9=0$$

28 olish uchun 19 va 9'ni qo'shing.

$$-48x^{2}-18x+28=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -48 ni a, -18 ni b va 28 ni c bilan almashtiring.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-48\right)\times 28}}{2\left(-48\right)}$$

-18 kvadratini chiqarish.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-48\right)\times 28}}{2\left(-48\right)}$$

-4 ni -48 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324+192\times 28}}{2\left(-48\right)}$$

192 ni 28 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324+5376}}{2\left(-48\right)}$$

324 ni 5376 ga qo'shish.

$$x=\frac{-\left(-18\right)±\sqrt{5700}}{2\left(-48\right)}$$

5700 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-18\right)±10\sqrt{57}}{2\left(-48\right)}$$

-18 ning teskarisi 18 ga teng.

$$x=\frac{18±10\sqrt{57}}{2\left(-48\right)}$$

2 ni -48 marotabaga ko'paytirish.

$$x=\frac{18±10\sqrt{57}}{-96}$$

x=\frac{18±10\sqrt{57}}{-96} tenglamasini yeching, bunda ± musbat. 18 ni 10\sqrt{57} ga qo'shish.

$$x=\frac{10\sqrt{57}+18}{-96}$$

18+10\sqrt{57} ni -96 ga bo'lish.

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

x=\frac{18±10\sqrt{57}}{-96} tenglamasini yeching, bunda ± manfiy. 18 dan 10\sqrt{57} ni ayirish.

$$x=\frac{18-10\sqrt{57}}{-96}$$

18-10\sqrt{57} ni -96 ga bo'lish.

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

Tenglama yechildi.

$$x=-\frac{5\sqrt{57}}{48}-\frac{3}{16} x=\frac{5\sqrt{57}}{48}-\frac{3}{16}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{1}{2},\frac{3}{4} qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(4x-3\right)\left(2x+1\right) ga, 2x+1,4x-3 ning eng kichik karralisiga ko‘paytiring.

$$\left(4x-3\right)\left(4x-3\right)-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

\left(4x-3\right)^{2} hosil qilish uchun 4x-3 va 4x-3 ni ko'paytirish.

$$\left(4x-3\right)^{2}-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-3\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=3\left(4x-3\right)\left(2x+1\right)$$

3 ga 4x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=\left(12x-9\right)\left(2x+1\right)$$

12x-9 ga 2x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)=24x^{2}-6x-9$$

Ikkala tarafdan 24x^{2} ni ayirish.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}=-6x-9$$

6x ni ikki tarafga qo’shing.

$$16x^{2}-24x+9-10\left(2x+1\right)\left(2x-1\right)-24x^{2}+6x=-9$$

-10 ga 2x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x^{2}-24x+9+\left(-20x-10\right)\left(2x-1\right)-24x^{2}+6x=-9$$

-20x-10 ga 2x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$16x^{2}-24x+9-40x^{2}+10-24x^{2}+6x=-9$$

-24x^{2} ni olish uchun 16x^{2} va -40x^{2} ni birlashtirish.

$$-24x^{2}-24x+9+10-24x^{2}+6x=-9$$

19 olish uchun 9 va 10'ni qo'shing.

$$-24x^{2}-24x+19-24x^{2}+6x=-9$$

-48x^{2} ni olish uchun -24x^{2} va -24x^{2} ni birlashtirish.

$$-48x^{2}-24x+19+6x=-9$$

-18x ni olish uchun -24x va 6x ni birlashtirish.

$$-48x^{2}-18x+19=-9$$

Ikkala tarafdan 19 ni ayirish.

$$-48x^{2}-18x=-9-19$$

-28 olish uchun -9 dan 19 ni ayirish.

$$-48x^{2}-18x=-28$$

Ikki tarafini -48 ga bo‘ling.

$$\frac{-48x^{2}-18x}{-48}=-\frac{28}{-48}$$

-48 ga bo'lish -48 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{18}{-48}\right)x=-\frac{28}{-48}$$

\frac{-18}{-48} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{3}{8}x=-\frac{28}{-48}$$

\frac{-28}{-48} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{3}{8}x=\frac{7}{12}$$

\frac{3}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{16} olish uchun. Keyin, \frac{3}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{8}x+\left(\frac{3}{16}\right)^{2}=\frac{7}{12}+\left(\frac{3}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{16} kvadratini chiqarish.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{7}{12}+\frac{9}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{12} ni \frac{9}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3}{8}x+\frac{9}{256}=\frac{475}{768}$$

x^{2}+\frac{3}{8}x+\frac{9}{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.

$$\left(x+\frac{3}{16}\right)^{2}=\frac{475}{768}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{16}\right)^{2}}=\sqrt{\frac{475}{768}}$$

Qisqartirish.

$$x+\frac{3}{16}=\frac{5\sqrt{57}}{48} x+\frac{3}{16}=-\frac{5\sqrt{57}}{48}$$

Tenglamaning ikkala tarafidan \frac{3}{16} ni ayirish.

$$x=\frac{5\sqrt{57}}{48}-\frac{3}{16} x=-\frac{5\sqrt{57}}{48}-\frac{3}{16}$$