x^2-(7+x/2)((7+x/2)+x)=11 eching

$${ x }^{ 2 } -( \frac{ 7+x }{ 2 } )(( \frac{ 7+x }{ 2 } )+x)=11$$

x uchun yechish

$$x=-3$$

$$x=31$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

7+x ga \frac{7+x}{2}+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

7\times \frac{7+x}{2} ni yagona kasrga aylantiring.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

x\times \frac{7+x}{2} ni yagona kasrga aylantiring.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

\frac{7\left(7+x\right)}{2} va \frac{x\left(7+x\right)}{2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

7\left(7+x\right)+x\left(7+x\right) ichidagi ko‘paytirishlarni bajaring.

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

49+7x+7x+x^{2} kabi iboralarga o‘xshab birlashtiring.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

\frac{49+14x+x^{2}}{2}+7x+x^{2} teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

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

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

\frac{49}{2}+7x+\frac{1}{2}x^{2} natijani olish uchun 49+14x+x^{2} ning har bir ifodasini 2 ga bo‘ling.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

\frac{49}{2}+7x+\frac{1}{2}x^{2} teskarisini topish uchun har birining teskarisini toping.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

\frac{1}{2}x^{2} ni olish uchun x^{2} va -\frac{1}{2}x^{2} ni birlashtirish.

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

-14x ni olish uchun -7x va -7x ni birlashtirish.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

Ikkala tarafdan 22 ni ayirish.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x-22=0$$

-\frac{93}{2} olish uchun -\frac{49}{2} dan 22 ni ayirish.

$$\frac{1}{2}x^{2}-\frac{93}{2}-14x=0$$

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.

$$\frac{1}{2}x^{2}-14x-\frac{93}{2}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{2} ni a, -14 ni b va -\frac{93}{2} ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times \frac{1}{2}\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

-4 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-2\left(-\frac{93}{2}\right)}}{2\times \frac{1}{2}}$$

-2 ni -\frac{93}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+93}}{2\times \frac{1}{2}}$$

196 ni 93 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{289}}{2\times \frac{1}{2}}$$

289 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±17}{2\times \frac{1}{2}}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±17}{2\times \frac{1}{2}}$$

2 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{14±17}{1}$$

x=\frac{14±17}{1} tenglamasini yeching, bunda ± musbat. 14 ni 17 ga qo'shish.

$$x=\frac{31}{1}$$

31 ni 1 ga bo'lish.

$$x=31$$

x=\frac{14±17}{1} tenglamasini yeching, bunda ± manfiy. 14 dan 17 ni ayirish.

$$x=-\frac{3}{1}$$

-3 ni 1 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=31 x=-3$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2x^{2}-\left(7+x\right)\left(\frac{7+x}{2}+x\right)=22$$

7+x ga \frac{7+x}{2}+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-\left(7\times \frac{7+x}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

7\times \frac{7+x}{2} ni yagona kasrga aylantiring.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+x\times \frac{7+x}{2}+x^{2}\right)=22$$

x\times \frac{7+x}{2} ni yagona kasrga aylantiring.

$$2x^{2}-\left(\frac{7\left(7+x\right)}{2}+7x+\frac{x\left(7+x\right)}{2}+x^{2}\right)=22$$

\frac{7\left(7+x\right)}{2} va \frac{x\left(7+x\right)}{2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$2x^{2}-\left(\frac{7\left(7+x\right)+x\left(7+x\right)}{2}+7x+x^{2}\right)=22$$

7\left(7+x\right)+x\left(7+x\right) ichidagi ko‘paytirishlarni bajaring.

$$2x^{2}-\left(\frac{49+7x+7x+x^{2}}{2}+7x+x^{2}\right)=22$$

49+7x+7x+x^{2} kabi iboralarga o‘xshab birlashtiring.

$$2x^{2}-\left(\frac{49+14x+x^{2}}{2}+7x+x^{2}\right)=22$$

\frac{49+14x+x^{2}}{2}+7x+x^{2} teskarisini topish uchun har birining teskarisini toping.

$$2x^{2}-\frac{49+14x+x^{2}}{2}-7x-x^{2}=22$$

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

$$x^{2}-\frac{49+14x+x^{2}}{2}-7x=22$$

\frac{49}{2}+7x+\frac{1}{2}x^{2} natijani olish uchun 49+14x+x^{2} ning har bir ifodasini 2 ga bo‘ling.

$$x^{2}-\left(\frac{49}{2}+7x+\frac{1}{2}x^{2}\right)-7x=22$$

\frac{49}{2}+7x+\frac{1}{2}x^{2} teskarisini topish uchun har birining teskarisini toping.

$$x^{2}-\frac{49}{2}-7x-\frac{1}{2}x^{2}-7x=22$$

\frac{1}{2}x^{2} ni olish uchun x^{2} va -\frac{1}{2}x^{2} ni birlashtirish.

$$\frac{1}{2}x^{2}-\frac{49}{2}-7x-7x=22$$

-14x ni olish uchun -7x va -7x ni birlashtirish.

$$\frac{1}{2}x^{2}-\frac{49}{2}-14x=22$$

\frac{49}{2} ni ikki tarafga qo’shing.

$$\frac{1}{2}x^{2}-14x=22+\frac{49}{2}$$

\frac{93}{2} olish uchun 22 va \frac{49}{2}'ni qo'shing.

$$\frac{1}{2}x^{2}-14x=\frac{93}{2}$$

Ikkala tarafini 2 ga ko‘paytiring.

$$\frac{\frac{1}{2}x^{2}-14x}{\frac{1}{2}}=\frac{\frac{93}{2}}{\frac{1}{2}}$$

\frac{1}{2} ga bo'lish \frac{1}{2} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{14}{\frac{1}{2}}\right)x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

-14 ni \frac{1}{2} ga bo'lish -14 ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-28x=\frac{\frac{93}{2}}{\frac{1}{2}}$$

\frac{93}{2} ni \frac{1}{2} ga bo'lish \frac{93}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-28x=93$$

-28 ni bo‘lish, x shartining koeffitsienti, 2 ga -14 olish uchun. Keyin, -14 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-28x+\left(-14\right)^{2}=93+\left(-14\right)^{2}$$

-14 kvadratini chiqarish.

$$x^{2}-28x+196=93+196$$

93 ni 196 ga qo'shish.

$$x^{2}-28x+196=289$$

x^{2}-28x+196 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-14\right)^{2}=289$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{289}$$

Qisqartirish.

$$x-14=17 x-14=-17$$

14 ni tenglamaning ikkala tarafiga qo'shish.

$$x=31 x=-3$$