1/2xleft(x-1right)=45 eching

$$\frac{ 1 }{ 2 } x \left( x-1 \right) = 45$$

x uchun yechish

$$x=10$$

$$x=-9$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$\frac{1}{2}xx+\frac{1}{2}x\left(-1\right)=45$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$\frac{1}{2}x^{2}+\frac{1}{2}x\left(-1\right)=45$$

-\frac{1}{2} hosil qilish uchun \frac{1}{2} va -1 ni ko'paytirish.

$$\frac{1}{2}x^{2}-\frac{1}{2}x=45$$

Ikkala tarafdan 45 ni ayirish.

$$\frac{1}{2}x^{2}-\frac{1}{2}x-45=0$$

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{1}{2} ni b va -45 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}-4\times \frac{1}{2}\left(-45\right)}}{2\times \frac{1}{2}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-4\times \frac{1}{2}\left(-45\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-2\left(-45\right)}}{2\times \frac{1}{2}}$$

-2 ni -45 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+90}}{2\times \frac{1}{2}}$$

\frac{1}{4} ni 90 ga qo'shish.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{361}{4}}}{2\times \frac{1}{2}}$$

\frac{361}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{19}{2}}{2\times \frac{1}{2}}$$

-\frac{1}{2} ning teskarisi \frac{1}{2} ga teng.

$$x=\frac{\frac{1}{2}±\frac{19}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{\frac{1}{2}±\frac{19}{2}}{1}$$

x=\frac{\frac{1}{2}±\frac{19}{2}}{1} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{19}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{10}{1}$$

10 ni 1 ga bo'lish.

$$x=10$$

x=\frac{\frac{1}{2}±\frac{19}{2}}{1} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{19}{2} ni \frac{1}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=-\frac{9}{1}$$

-9 ni 1 ga bo'lish.

$$x=-9$$

Tenglama yechildi.

$$x=10 x=-9$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\frac{1}{2}xx+\frac{1}{2}x\left(-1\right)=45$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$\frac{1}{2}x^{2}+\frac{1}{2}x\left(-1\right)=45$$

-\frac{1}{2} hosil qilish uchun \frac{1}{2} va -1 ni ko'paytirish.

$$\frac{1}{2}x^{2}-\frac{1}{2}x=45$$

Ikkala tarafini 2 ga ko‘paytiring.

$$\frac{\frac{1}{2}x^{2}-\frac{1}{2}x}{\frac{1}{2}}=\frac{45}{\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{\frac{1}{2}}{\frac{1}{2}}\right)x=\frac{45}{\frac{1}{2}}$$

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

$$x^{2}-x=\frac{45}{\frac{1}{2}}$$

45 ni \frac{1}{2} ga bo'lish 45 ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-x=90$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=90+\left(-\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$x^{2}-x+\frac{1}{4}=90+\frac{1}{4}$$

90 ni \frac{1}{4} ga qo'shish.

$$x^{2}-x+\frac{1}{4}=\frac{361}{4}$$

x^{2}-x+\frac{1}{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.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{361}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{19}{2} x-\frac{1}{2}=-\frac{19}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=-9$$