2x-3*5/4x+3=1 eching

$$2x-3 \times \frac{ 5 }{ 4x+3 } =1$$

x uchun yechish

$$x = \frac{\sqrt{145} - 1}{8} \approx 1,380199322$$

$$x=\frac{-\sqrt{145}-1}{8}\approx -1,630199322$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{3}{4} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4x+3 ga ko'paytirish.

$$2x\left(4x+3\right)-3\times 5=4x+3$$

2x ga 4x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}+6x-3\times 5=4x+3$$

15 hosil qilish uchun 3 va 5 ni ko'paytirish.

$$8x^{2}+6x-15=4x+3$$

Ikkala tarafdan 4x ni ayirish.

$$8x^{2}+6x-15-4x=3$$

2x ni olish uchun 6x va -4x ni birlashtirish.

$$8x^{2}+2x-15=3$$

Ikkala tarafdan 3 ni ayirish.

$$8x^{2}+2x-15-3=0$$

-18 olish uchun -15 dan 3 ni ayirish.

$$8x^{2}+2x-18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 2 ni b va -18 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 8\left(-18\right)}}{2\times 8}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 8\left(-18\right)}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-32\left(-18\right)}}{2\times 8}$$

-32 ni -18 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+576}}{2\times 8}$$

4 ni 576 ga qo'shish.

$$x=\frac{-2±\sqrt{580}}{2\times 8}$$

580 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{145}}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{145}}{16}$$

x=\frac{-2±2\sqrt{145}}{16} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{145} ga qo'shish.

$$x=\frac{2\sqrt{145}-2}{16}$$

-2+2\sqrt{145} ni 16 ga bo'lish.

$$x=\frac{\sqrt{145}-1}{8}$$

x=\frac{-2±2\sqrt{145}}{16} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{145} ni ayirish.

$$x=\frac{-2\sqrt{145}-2}{16}$$

-2-2\sqrt{145} ni 16 ga bo'lish.

$$x=\frac{-\sqrt{145}-1}{8}$$

Tenglama yechildi.

$$x=\frac{\sqrt{145}-1}{8} x=\frac{-\sqrt{145}-1}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{3}{4} teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4x+3 ga ko'paytirish.

$$2x\left(4x+3\right)-3\times 5=4x+3$$

2x ga 4x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$8x^{2}+6x-3\times 5=4x+3$$

15 hosil qilish uchun 3 va 5 ni ko'paytirish.

$$8x^{2}+6x-15=4x+3$$

Ikkala tarafdan 4x ni ayirish.

$$8x^{2}+6x-15-4x=3$$

2x ni olish uchun 6x va -4x ni birlashtirish.

$$8x^{2}+2x-15=3$$

15 ni ikki tarafga qo’shing.

$$8x^{2}+2x=3+15$$

18 olish uchun 3 va 15'ni qo'shing.

$$8x^{2}+2x=18$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}+2x}{8}=\frac{18}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{8}x=\frac{18}{8}$$

\frac{2}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{1}{4}x=\frac{18}{8}$$

\frac{18}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{1}{4}x=\frac{9}{4}$$

\frac{1}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{8} olish uchun. Keyin, \frac{1}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{9}{4}+\left(\frac{1}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{8} kvadratini chiqarish.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{9}{4}+\frac{1}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{4} ni \frac{1}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{145}{64}$$

x^{2}+\frac{1}{4}x+\frac{1}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{8}\right)^{2}=\frac{145}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{\frac{145}{64}}$$

Qisqartirish.

$$x+\frac{1}{8}=\frac{\sqrt{145}}{8} x+\frac{1}{8}=-\frac{\sqrt{145}}{8}$$

Tenglamaning ikkala tarafidan \frac{1}{8} ni ayirish.

$$x=\frac{\sqrt{145}-1}{8} x=\frac{-\sqrt{145}-1}{8}$$