112x^2-7x-9=0 eching

$$112 x ^ { 2 } - 7 x - 9 = 0$$

x uchun yechish

$$x=\frac{\sqrt{4081}}{224}+\frac{1}{32}\approx 0,316440647$$

$$x=-\frac{\sqrt{4081}}{224}+\frac{1}{32}\approx -0,253940647$$

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.

$$112x^{2}-7x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 112 ni a, -7 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 112\left(-9\right)}}{2\times 112}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 112\left(-9\right)}}{2\times 112}$$

-4 ni 112 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-448\left(-9\right)}}{2\times 112}$$

-448 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+4032}}{2\times 112}$$

49 ni 4032 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{4081}}{2\times 112}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{4081}}{2\times 112}$$

2 ni 112 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{4081}}{224}$$

x=\frac{7±\sqrt{4081}}{224} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{4081} ga qo'shish.

$$x=\frac{\sqrt{4081}+7}{224}$$

7+\sqrt{4081} ni 224 ga bo'lish.

$$x=\frac{\sqrt{4081}}{224}+\frac{1}{32}$$

x=\frac{7±\sqrt{4081}}{224} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{4081} ni ayirish.

$$x=\frac{7-\sqrt{4081}}{224}$$

7-\sqrt{4081} ni 224 ga bo'lish.

$$x=-\frac{\sqrt{4081}}{224}+\frac{1}{32}$$

Tenglama yechildi.

$$x=\frac{\sqrt{4081}}{224}+\frac{1}{32} x=-\frac{\sqrt{4081}}{224}+\frac{1}{32}$$

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.

$$112x^{2}-7x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$112x^{2}-7x-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$112x^{2}-7x=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$112x^{2}-7x=9$$

Ikki tarafini 112 ga bo‘ling.

$$\frac{112x^{2}-7x}{112}=\frac{9}{112}$$

112 ga bo'lish 112 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{7}{112}\right)x=\frac{9}{112}$$

\frac{-7}{112} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{16}x=\frac{9}{112}$$

-\frac{1}{16} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{32} olish uchun. Keyin, -\frac{1}{32} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{16}x+\left(-\frac{1}{32}\right)^{2}=\frac{9}{112}+\left(-\frac{1}{32}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{32} kvadratini chiqarish.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=\frac{9}{112}+\frac{1}{1024}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{112} ni \frac{1}{1024} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=\frac{583}{7168}$$

x^{2}-\frac{1}{16}x+\frac{1}{1024} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{32}\right)^{2}=\frac{583}{7168}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{32}\right)^{2}}=\sqrt{\frac{583}{7168}}$$

Qisqartirish.

$$x-\frac{1}{32}=\frac{\sqrt{4081}}{224} x-\frac{1}{32}=-\frac{\sqrt{4081}}{224}$$

\frac{1}{32} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{4081}}{224}+\frac{1}{32} x=-\frac{\sqrt{4081}}{224}+\frac{1}{32}$$