-1/8x^2+1/2x+4=0 eching

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

x uchun yechish

$$x=-4$$

$$x=8$$

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.

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

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{1}{2} ni b va 4 ni c bilan almashtiring.

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

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

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

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

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

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

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

\frac{1}{4} ni 2 ga qo'shish.

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

\frac{9}{4} ning kvadrat ildizini chiqarish.

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

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

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

x=\frac{-\frac{1}{2}±\frac{3}{2}}{-\frac{1}{4}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{3}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

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

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

$$x=-4$$

x=\frac{-\frac{1}{2}±\frac{3}{2}}{-\frac{1}{4}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{3}{2} ni -\frac{1}{2} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

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

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

$$x=8$$

Tenglama yechildi.

$$x=-4 x=8$$

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.

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

Tenglamaning ikkala tarafidan 4 ni ayirish.

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

O‘zidan 4 ayirilsa 0 qoladi.

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

Ikkala tarafini -8 ga ko‘paytiring.

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

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

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

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

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

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

$$x^{2}-4x=32$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=32+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=32+4$$

32 ni 4 ga qo'shish.

$$x^{2}-4x+4=36$$

x^{2}-4x+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.

$$\left(x-2\right)^{2}=36$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{36}$$

Qisqartirish.

$$x-2=6 x-2=-6$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=-4$$