(1/2(2x+2)(x+4))-(1/2(x+1)(x))=60 eching

$$( \frac{ 1 }{ 2 } (2x+2)(x+4))-( \frac{ 1 }{ 2 } (x+1)(x))=60$$

x uchun yechish

$$x=-16$$

$$x=7$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{1}{2} ga 2x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

x+1 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

Ikkala tarafdan 60 ni ayirish.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x-60=0$$

-\frac{1}{2} ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x-60=0$$

-\frac{1}{2}x-\frac{1}{2} ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x-60=0$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x-60=0$$

\frac{9}{2}x ni olish uchun 5x va -\frac{1}{2}x ni birlashtirish.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4-60=0$$

-56 olish uchun 4 dan 60 ni ayirish.

$$\frac{1}{2}x^{2}+\frac{9}{2}x-56=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{2} ni a, \frac{9}{2} ni b va -56 ni c bilan almashtiring.

$$x=\frac{-\frac{9}{2}±\sqrt{\left(\frac{9}{2}\right)^{2}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{2} kvadratini chiqarish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-4\times \frac{1}{2}\left(-56\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}-2\left(-56\right)}}{2\times \frac{1}{2}}$$

-2 ni -56 marotabaga ko'paytirish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{81}{4}+112}}{2\times \frac{1}{2}}$$

\frac{81}{4} ni 112 ga qo'shish.

$$x=\frac{-\frac{9}{2}±\sqrt{\frac{529}{4}}}{2\times \frac{1}{2}}$$

\frac{529}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{9}{2}±\frac{23}{2}}{1}$$

x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{2} ni \frac{23}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{7}{1}$$

7 ni 1 ga bo'lish.

$$x=7$$

x=\frac{-\frac{9}{2}±\frac{23}{2}}{1} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{23}{2} ni -\frac{9}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=-\frac{16}{1}$$

-16 ni 1 ga bo'lish.

$$x=-16$$

Tenglama yechildi.

$$x=7 x=-16$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{1}{2} ga 2x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(x+1\right)\left(x+4\right)-\frac{1}{2}\left(x+1\right)x=60$$

x+1 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+5x+4-\frac{1}{2}\left(x+1\right)x=60$$

-\frac{1}{2} ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+5x+4+\left(-\frac{1}{2}x-\frac{1}{2}\right)x=60$$

-\frac{1}{2}x-\frac{1}{2} ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+5x+4-\frac{1}{2}x^{2}-\frac{1}{2}x=60$$

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

$$\frac{1}{2}x^{2}+5x+4-\frac{1}{2}x=60$$

\frac{9}{2}x ni olish uchun 5x va -\frac{1}{2}x ni birlashtirish.

$$\frac{1}{2}x^{2}+\frac{9}{2}x+4=60$$

Ikkala tarafdan 4 ni ayirish.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=60-4$$

56 olish uchun 60 dan 4 ni ayirish.

$$\frac{1}{2}x^{2}+\frac{9}{2}x=56$$

Ikkala tarafini 2 ga ko‘paytiring.

$$\frac{\frac{1}{2}x^{2}+\frac{9}{2}x}{\frac{1}{2}}=\frac{56}{\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{9}{2}}{\frac{1}{2}}x=\frac{56}{\frac{1}{2}}$$

\frac{9}{2} ni \frac{1}{2} ga bo'lish \frac{9}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}+9x=\frac{56}{\frac{1}{2}}$$

56 ni \frac{1}{2} ga bo'lish 56 ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}+9x=112$$

9 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{2} olish uchun. Keyin, \frac{9}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=112+\left(\frac{9}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{2} kvadratini chiqarish.

$$x^{2}+9x+\frac{81}{4}=112+\frac{81}{4}$$

112 ni \frac{81}{4} ga qo'shish.

$$x^{2}+9x+\frac{81}{4}=\frac{529}{4}$$

x^{2}+9x+\frac{81}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{529}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Qisqartirish.

$$x+\frac{9}{2}=\frac{23}{2} x+\frac{9}{2}=-\frac{23}{2}$$

Tenglamaning ikkala tarafidan \frac{9}{2} ni ayirish.

$$x=7 x=-16$$