1/8x^2-3/4x=2 eching

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

x uchun yechish

$$x=-2$$

$$x=8$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

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

Tenglamaning ikkala tarafidan 2 ni ayirish.

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

O‘zidan 2 ayirilsa 0 qoladi.

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

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 -2 ni c bilan almashtiring.

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

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

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

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

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

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

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

\frac{9}{16} ni 1 ga qo'shish.

$$x=\frac{-\left(-\frac{3}{4}\right)±\sqrt{\frac{25}{16}}}{2\times \frac{1}{8}}$$

\frac{25}{16} ning kvadrat ildizini chiqarish.

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

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

$$x=\frac{\frac{3}{4}±\frac{5}{4}}{2\times \frac{1}{8}}$$

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

$$x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}}$$

x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{4} ni \frac{5}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

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

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

$$x=8$$

x=\frac{\frac{3}{4}±\frac{5}{4}}{\frac{1}{4}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{5}{4} ni \frac{3}{4} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

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

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

$$x=-2$$

Tenglama yechildi.

$$x=8 x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

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

Ikkala tarafini 8 ga ko‘paytiring.

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

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

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

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

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

2 ni \frac{1}{8} ga bo'lish 2 ga k'paytirish \frac{1}{8} ga qaytarish.

$$x^{2}-6x=16$$

-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.

$$x^{2}-6x+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=16+9$$

16 ni 9 ga qo'shish.

$$x^{2}-6x+9=25$$

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.

$$\left(x-3\right)^{2}=25$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{25}$$

Qisqartirish.

$$x-3=5 x-3=-5$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=-2$$