-8-1/8x^2+x=(1/4x-1)(3-x) eching

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

x uchun yechish

$$x=-4$$

$$x=10$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{1}{4}x-1 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-8-\frac{1}{8}x^{2}+x=\frac{7}{4}x-\frac{1}{4}x^{2}-3$$

Ikkala tarafdan \frac{7}{4}x ni ayirish.

$$-8-\frac{1}{8}x^{2}+x-\frac{7}{4}x=-\frac{1}{4}x^{2}-3$$

-\frac{3}{4}x ni olish uchun x va -\frac{7}{4}x ni birlashtirish.

$$-8-\frac{1}{8}x^{2}-\frac{3}{4}x=-\frac{1}{4}x^{2}-3$$

\frac{1}{4}x^{2} ni ikki tarafga qo’shing.

$$-8-\frac{1}{8}x^{2}-\frac{3}{4}x+\frac{1}{4}x^{2}=-3$$

\frac{1}{8}x^{2} ni olish uchun -\frac{1}{8}x^{2} va \frac{1}{4}x^{2} ni birlashtirish.

$$-8+\frac{1}{8}x^{2}-\frac{3}{4}x=-3$$

3 ni ikki tarafga qo’shing.

$$-8+\frac{1}{8}x^{2}-\frac{3}{4}x+3=0$$

-5 olish uchun -8 va 3'ni qo'shing.

$$-5+\frac{1}{8}x^{2}-\frac{3}{4}x=0$$

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.

$$\frac{1}{8}x^{2}-\frac{3}{4}x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{8} ni a, -\frac{3}{4} ni b va -5 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\left(-\frac{3}{4}\right)^{2}-4\times \frac{1}{8}\left(-5\right)}}{2\times \frac{1}{8}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-4\times \frac{1}{8}\left(-5\right)}}{2\times \frac{1}{8}}$$

-4 ni \frac{1}{8} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}-\frac{1}{2}\left(-5\right)}}{2\times \frac{1}{8}}$$

-\frac{1}{2} ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{9}{16}+\frac{5}{2}}}{2\times \frac{1}{8}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{16} ni \frac{5}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{49}{16}}}{2\times \frac{1}{8}}$$

\frac{49}{16} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{4}\right)±\frac{7}{4}}{2\times \frac{1}{8}}$$

-\frac{3}{4} ning teskarisi \frac{3}{4} ga teng.

$$x=\frac{\frac{3}{4}±\frac{7}{4}}{2\times \frac{1}{8}}$$

2 ni \frac{1}{8} marotabaga ko'paytirish.

$$x=\frac{\frac{3}{4}±\frac{7}{4}}{\frac{1}{4}}$$

x=\frac{\frac{3}{4}±\frac{7}{4}}{\frac{1}{4}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{4} ni \frac{7}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{\frac{5}{2}}{\frac{1}{4}}$$

\frac{5}{2} ni \frac{1}{4} ga bo'lish \frac{5}{2} ga k'paytirish \frac{1}{4} ga qaytarish.

$$x=10$$

x=\frac{\frac{3}{4}±\frac{7}{4}}{\frac{1}{4}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{7}{4} ni \frac{3}{4} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$x=-\frac{1}{\frac{1}{4}}$$

-1 ni \frac{1}{4} ga bo'lish -1 ga k'paytirish \frac{1}{4} ga qaytarish.

$$x=-4$$

Tenglama yechildi.

$$x=10 x=-4$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{1}{4}x-1 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-8-\frac{1}{8}x^{2}+x=\frac{7}{4}x-\frac{1}{4}x^{2}-3$$

Ikkala tarafdan \frac{7}{4}x ni ayirish.

$$-8-\frac{1}{8}x^{2}+x-\frac{7}{4}x=-\frac{1}{4}x^{2}-3$$

-\frac{3}{4}x ni olish uchun x va -\frac{7}{4}x ni birlashtirish.

$$-8-\frac{1}{8}x^{2}-\frac{3}{4}x=-\frac{1}{4}x^{2}-3$$

\frac{1}{4}x^{2} ni ikki tarafga qo’shing.

$$-8-\frac{1}{8}x^{2}-\frac{3}{4}x+\frac{1}{4}x^{2}=-3$$

\frac{1}{8}x^{2} ni olish uchun -\frac{1}{8}x^{2} va \frac{1}{4}x^{2} ni birlashtirish.

$$-8+\frac{1}{8}x^{2}-\frac{3}{4}x=-3$$

8 ni ikki tarafga qo’shing.

$$\frac{1}{8}x^{2}-\frac{3}{4}x=-3+8$$

5 olish uchun -3 va 8'ni qo'shing.

$$\frac{1}{8}x^{2}-\frac{3}{4}x=5$$

Ikkala tarafini 8 ga ko‘paytiring.

$$\frac{\frac{1}{8}x^{2}-\frac{3}{4}x}{\frac{1}{8}}=\frac{5}{\frac{1}{8}}$$

\frac{1}{8} ga bo'lish \frac{1}{8} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{3}{4}}{\frac{1}{8}}\right)x=\frac{5}{\frac{1}{8}}$$

-\frac{3}{4} ni \frac{1}{8} ga bo'lish -\frac{3}{4} ga k'paytirish \frac{1}{8} ga qaytarish.

$$x^{2}-6x=\frac{5}{\frac{1}{8}}$$

5 ni \frac{1}{8} ga bo'lish 5 ga k'paytirish \frac{1}{8} ga qaytarish.

$$x^{2}-6x=40$$

-6 ni bo‘lish, x shartining koeffitsienti, 2 ga -3 olish uchun. Keyin, -3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-6x+\left(-3\right)^{2}=40+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=40+9$$

40 ni 9 ga qo'shish.

$$x^{2}-6x+9=49$$

x^{2}-6x+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-3\right)^{2}=49$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{49}$$

Qisqartirish.

$$x-3=7 x-3=-7$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=-4$$