2*(3x+4)-2*(5x+2)=3:(2x+2)+4*(4x+10) eching

$$2 \cdot ( 3 x + 4 ) - 2 \cdot ( 5 x + 2 ) = 3 : ( 2 x + 2 ) + 4 \cdot ( 4 x + 10 )$$

x uchun yechish

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1,108452405$$

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}\approx -1,691547595$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x+1\right) ga ko'paytirish.

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

4 hosil qilish uchun 2 va 2 ni ko'paytirish.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

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

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

12x+16 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-4 hosil qilish uchun -2 va 2 ni ko'paytirish.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-4 ga 5x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-20x-8 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-8x^{2} ni olish uchun 12x^{2} va -20x^{2} ni birlashtirish.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

0 ni olish uchun 28x va -28x ni birlashtirish.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

8 olish uchun 16 dan 8 ni ayirish.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

8 ga 4x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

32x+80 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

83 olish uchun 3 va 80'ni qo'shing.

$$-8x^{2}+8=83+32x^{2}+112x$$

Ikkala tarafdan 83 ni ayirish.

$$-8x^{2}+8-83=32x^{2}+112x$$

-75 olish uchun 8 dan 83 ni ayirish.

$$-8x^{2}-75=32x^{2}+112x$$

Ikkala tarafdan 32x^{2} ni ayirish.

$$-8x^{2}-75-32x^{2}=112x$$

-40x^{2} ni olish uchun -8x^{2} va -32x^{2} ni birlashtirish.

$$-40x^{2}-75=112x$$

Ikkala tarafdan 112x ni ayirish.

$$-40x^{2}-75-112x=0$$

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.

$$-40x^{2}-112x-75=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -40 ni a, -112 ni b va -75 ni c bilan almashtiring.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

-112 kvadratini chiqarish.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\left(-40\right)\left(-75\right)}}{2\left(-40\right)}$$

-4 ni -40 marotabaga ko'paytirish.

$$x=\frac{-\left(-112\right)±\sqrt{12544+160\left(-75\right)}}{2\left(-40\right)}$$

160 ni -75 marotabaga ko'paytirish.

$$x=\frac{-\left(-112\right)±\sqrt{12544-12000}}{2\left(-40\right)}$$

12544 ni -12000 ga qo'shish.

$$x=\frac{-\left(-112\right)±\sqrt{544}}{2\left(-40\right)}$$

544 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-112\right)±4\sqrt{34}}{2\left(-40\right)}$$

-112 ning teskarisi 112 ga teng.

$$x=\frac{112±4\sqrt{34}}{2\left(-40\right)}$$

2 ni -40 marotabaga ko'paytirish.

$$x=\frac{112±4\sqrt{34}}{-80}$$

x=\frac{112±4\sqrt{34}}{-80} tenglamasini yeching, bunda ± musbat. 112 ni 4\sqrt{34} ga qo'shish.

$$x=\frac{4\sqrt{34}+112}{-80}$$

112+4\sqrt{34} ni -80 ga bo'lish.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$

x=\frac{112±4\sqrt{34}}{-80} tenglamasini yeching, bunda ± manfiy. 112 dan 4\sqrt{34} ni ayirish.

$$x=\frac{112-4\sqrt{34}}{-80}$$

112-4\sqrt{34} ni -80 ga bo'lish.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Tenglama yechildi.

$$x=-\frac{\sqrt{34}}{20}-\frac{7}{5} x=\frac{\sqrt{34}}{20}-\frac{7}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2\left(x+1\right) ga ko'paytirish.

$$2\left(3x+4\right)\times 2\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

4 hosil qilish uchun 2 va 2 ni ko'paytirish.

$$4\left(3x+4\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

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

$$\left(12x+16\right)\left(x+1\right)-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

12x+16 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$12x^{2}+28x+16-2\left(5x+2\right)\times 2\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-4 hosil qilish uchun -2 va 2 ni ko'paytirish.

$$12x^{2}+28x+16-4\left(5x+2\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-4 ga 5x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12x^{2}+28x+16+\left(-20x-8\right)\left(x+1\right)=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-20x-8 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$12x^{2}+28x+16-20x^{2}-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

-8x^{2} ni olish uchun 12x^{2} va -20x^{2} ni birlashtirish.

$$-8x^{2}+28x+16-28x-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

0 ni olish uchun 28x va -28x ni birlashtirish.

$$-8x^{2}+16-8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

8 olish uchun 16 dan 8 ni ayirish.

$$-8x^{2}+8=3+4\left(4x+10\right)\times 2\left(x+1\right)$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$-8x^{2}+8=3+8\left(4x+10\right)\left(x+1\right)$$

8 ga 4x+10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-8x^{2}+8=3+\left(32x+80\right)\left(x+1\right)$$

32x+80 ga x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-8x^{2}+8=3+32x^{2}+112x+80$$

83 olish uchun 3 va 80'ni qo'shing.

$$-8x^{2}+8=83+32x^{2}+112x$$

Ikkala tarafdan 32x^{2} ni ayirish.

$$-8x^{2}+8-32x^{2}=83+112x$$

-40x^{2} ni olish uchun -8x^{2} va -32x^{2} ni birlashtirish.

$$-40x^{2}+8=83+112x$$

Ikkala tarafdan 112x ni ayirish.

$$-40x^{2}+8-112x=83$$

Ikkala tarafdan 8 ni ayirish.

$$-40x^{2}-112x=83-8$$

75 olish uchun 83 dan 8 ni ayirish.

$$-40x^{2}-112x=75$$

Ikki tarafini -40 ga bo‘ling.

$$\frac{-40x^{2}-112x}{-40}=\frac{75}{-40}$$

-40 ga bo'lish -40 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{112}{-40}\right)x=\frac{75}{-40}$$

\frac{-112}{-40} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{14}{5}x=\frac{75}{-40}$$

\frac{75}{-40} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{14}{5}x=-\frac{15}{8}$$

\frac{14}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{5} olish uchun. Keyin, \frac{7}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{14}{5}x+\left(\frac{7}{5}\right)^{2}=-\frac{15}{8}+\left(\frac{7}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{5} kvadratini chiqarish.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=-\frac{15}{8}+\frac{49}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{15}{8} ni \frac{49}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{14}{5}x+\frac{49}{25}=\frac{17}{200}$$

x^{2}+\frac{14}{5}x+\frac{49}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{5}\right)^{2}=\frac{17}{200}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{5}\right)^{2}}=\sqrt{\frac{17}{200}}$$

Qisqartirish.

$$x+\frac{7}{5}=\frac{\sqrt{34}}{20} x+\frac{7}{5}=-\frac{\sqrt{34}}{20}$$

Tenglamaning ikkala tarafidan \frac{7}{5} ni ayirish.

$$x=\frac{\sqrt{34}}{20}-\frac{7}{5} x=-\frac{\sqrt{34}}{20}-\frac{7}{5}$$