13x^2-5x-20=0 eching

$$13 x ^ { 2 } - 5 x - 20 = 0$$

x uchun yechish

$$x = \frac{\sqrt{1065} + 5}{26} \approx 1,447474529$$

$$x=\frac{5-\sqrt{1065}}{26}\approx -1,062859144$$

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.

$$13x^{2}-5x-20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 13 ni a, -5 ni b va -20 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 13\left(-20\right)}}{2\times 13}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 13\left(-20\right)}}{2\times 13}$$

-4 ni 13 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-52\left(-20\right)}}{2\times 13}$$

-52 ni -20 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+1040}}{2\times 13}$$

25 ni 1040 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{1065}}{2\times 13}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±\sqrt{1065}}{2\times 13}$$

2 ni 13 marotabaga ko'paytirish.

$$x=\frac{5±\sqrt{1065}}{26}$$

x=\frac{5±\sqrt{1065}}{26} tenglamasini yeching, bunda ± musbat. 5 ni \sqrt{1065} ga qo'shish.

$$x=\frac{\sqrt{1065}+5}{26}$$

x=\frac{5±\sqrt{1065}}{26} tenglamasini yeching, bunda ± manfiy. 5 dan \sqrt{1065} ni ayirish.

$$x=\frac{5-\sqrt{1065}}{26}$$

Tenglama yechildi.

$$x=\frac{\sqrt{1065}+5}{26} x=\frac{5-\sqrt{1065}}{26}$$

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.

$$13x^{2}-5x-20=0$$

20 ni tenglamaning ikkala tarafiga qo'shish.

$$13x^{2}-5x-20-\left(-20\right)=-\left(-20\right)$$

O‘zidan -20 ayirilsa 0 qoladi.

$$13x^{2}-5x=-\left(-20\right)$$

0 dan -20 ni ayirish.

$$13x^{2}-5x=20$$

Ikki tarafini 13 ga bo‘ling.

$$\frac{13x^{2}-5x}{13}=\frac{20}{13}$$

13 ga bo'lish 13 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{5}{13}x=\frac{20}{13}$$

-\frac{5}{13} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{26} olish uchun. Keyin, -\frac{5}{26} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{5}{13}x+\left(-\frac{5}{26}\right)^{2}=\frac{20}{13}+\left(-\frac{5}{26}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{26} kvadratini chiqarish.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=\frac{20}{13}+\frac{25}{676}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{20}{13} ni \frac{25}{676} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=\frac{1065}{676}$$

x^{2}-\frac{5}{13}x+\frac{25}{676} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{5}{26}\right)^{2}=\frac{1065}{676}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{26}\right)^{2}}=\sqrt{\frac{1065}{676}}$$

Qisqartirish.

$$x-\frac{5}{26}=\frac{\sqrt{1065}}{26} x-\frac{5}{26}=-\frac{\sqrt{1065}}{26}$$

\frac{5}{26} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{1065}+5}{26} x=\frac{5-\sqrt{1065}}{26}$$