28x^2-7x-1=0 eching

$$28 { x }^{ 2 } -7x-1=0$$

x uchun yechish

$$x=\frac{\sqrt{161}}{56}+\frac{1}{8}\approx 0,351581742$$

$$x=-\frac{\sqrt{161}}{56}+\frac{1}{8}\approx -0,101581742$$

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.

$$28x^{2}-7x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 28 ni a, -7 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 28\left(-1\right)}}{2\times 28}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 28\left(-1\right)}}{2\times 28}$$

-4 ni 28 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-112\left(-1\right)}}{2\times 28}$$

-112 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+112}}{2\times 28}$$

49 ni 112 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{161}}{2\times 28}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{161}}{2\times 28}$$

2 ni 28 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{161}}{56}$$

x=\frac{7±\sqrt{161}}{56} tenglamasini yeching, bunda ± musbat. 7 ni \sqrt{161} ga qo'shish.

$$x=\frac{\sqrt{161}+7}{56}$$

7+\sqrt{161} ni 56 ga bo'lish.

$$x=\frac{\sqrt{161}}{56}+\frac{1}{8}$$

x=\frac{7±\sqrt{161}}{56} tenglamasini yeching, bunda ± manfiy. 7 dan \sqrt{161} ni ayirish.

$$x=\frac{7-\sqrt{161}}{56}$$

7-\sqrt{161} ni 56 ga bo'lish.

$$x=-\frac{\sqrt{161}}{56}+\frac{1}{8}$$

Tenglama yechildi.

$$x=\frac{\sqrt{161}}{56}+\frac{1}{8} x=-\frac{\sqrt{161}}{56}+\frac{1}{8}$$

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.

$$28x^{2}-7x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$28x^{2}-7x-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$28x^{2}-7x=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$28x^{2}-7x=1$$

Ikki tarafini 28 ga bo‘ling.

$$\frac{28x^{2}-7x}{28}=\frac{1}{28}$$

28 ga bo'lish 28 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{7}{28}\right)x=\frac{1}{28}$$

\frac{-7}{28} ulushini 7 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{4}x=\frac{1}{28}$$

-\frac{1}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{8} olish uchun. Keyin, -\frac{1}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\frac{1}{28}+\left(-\frac{1}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{8} kvadratini chiqarish.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{1}{28}+\frac{1}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{28} ni \frac{1}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{23}{448}$$

x^{2}-\frac{1}{4}x+\frac{1}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{23}{448}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{23}{448}}$$

Qisqartirish.

$$x-\frac{1}{8}=\frac{\sqrt{161}}{56} x-\frac{1}{8}=-\frac{\sqrt{161}}{56}$$

\frac{1}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{161}}{56}+\frac{1}{8} x=-\frac{\sqrt{161}}{56}+\frac{1}{8}$$