125x^2-78*5x+36125=0 eching

$$125 x ^ { 2 } - 78 \cdot 5 x + 36125 = 0$$

x uchun yechish (complex solution)

$$x=\frac{39+4\sqrt{11194}i}{25}\approx 1,56+16,92827221i$$

$$x=\frac{-4\sqrt{11194}i+39}{25}\approx 1,56-16,92827221i$$

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.

$$125x^{2}-390x+36125=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 125 ni a, -390 ni b va 36125 ni c bilan almashtiring.

$$x=\frac{-\left(-390\right)±\sqrt{\left(-390\right)^{2}-4\times 125\times 36125}}{2\times 125}$$

-390 kvadratini chiqarish.

$$x=\frac{-\left(-390\right)±\sqrt{152100-4\times 125\times 36125}}{2\times 125}$$

-4 ni 125 marotabaga ko'paytirish.

$$x=\frac{-\left(-390\right)±\sqrt{152100-500\times 36125}}{2\times 125}$$

-500 ni 36125 marotabaga ko'paytirish.

$$x=\frac{-\left(-390\right)±\sqrt{152100-18062500}}{2\times 125}$$

152100 ni -18062500 ga qo'shish.

$$x=\frac{-\left(-390\right)±\sqrt{-17910400}}{2\times 125}$$

-17910400 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-390\right)±40\sqrt{11194}i}{2\times 125}$$

-390 ning teskarisi 390 ga teng.

$$x=\frac{390±40\sqrt{11194}i}{2\times 125}$$

2 ni 125 marotabaga ko'paytirish.

$$x=\frac{390±40\sqrt{11194}i}{250}$$

x=\frac{390±40\sqrt{11194}i}{250} tenglamasini yeching, bunda ± musbat. 390 ni 40i\sqrt{11194} ga qo'shish.

$$x=\frac{390+40\sqrt{11194}i}{250}$$

390+40i\sqrt{11194} ni 250 ga bo'lish.

$$x=\frac{39+4\sqrt{11194}i}{25}$$

x=\frac{390±40\sqrt{11194}i}{250} tenglamasini yeching, bunda ± manfiy. 390 dan 40i\sqrt{11194} ni ayirish.

$$x=\frac{-40\sqrt{11194}i+390}{250}$$

390-40i\sqrt{11194} ni 250 ga bo'lish.

$$x=\frac{-4\sqrt{11194}i+39}{25}$$

Tenglama yechildi.

$$x=\frac{39+4\sqrt{11194}i}{25} x=\frac{-4\sqrt{11194}i+39}{25}$$

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.

$$125x^{2}-390x+36125=0$$

Tenglamaning ikkala tarafidan 36125 ni ayirish.

$$125x^{2}-390x+36125-36125=-36125$$

O‘zidan 36125 ayirilsa 0 qoladi.

$$125x^{2}-390x=-36125$$

Ikki tarafini 125 ga bo‘ling.

$$\frac{125x^{2}-390x}{125}=-\frac{36125}{125}$$

125 ga bo'lish 125 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{390}{125}\right)x=-\frac{36125}{125}$$

\frac{-390}{125} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{78}{25}x=-\frac{36125}{125}$$

-36125 ni 125 ga bo'lish.

$$x^{2}-\frac{78}{25}x=-289$$

-\frac{78}{25} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{39}{25} olish uchun. Keyin, -\frac{39}{25} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{78}{25}x+\left(-\frac{39}{25}\right)^{2}=-289+\left(-\frac{39}{25}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{39}{25} kvadratini chiqarish.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=-289+\frac{1521}{625}$$

-289 ni \frac{1521}{625} ga qo'shish.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=-\frac{179104}{625}$$

x^{2}-\frac{78}{25}x+\frac{1521}{625} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{39}{25}\right)^{2}=-\frac{179104}{625}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{39}{25}\right)^{2}}=\sqrt{-\frac{179104}{625}}$$

Qisqartirish.

$$x-\frac{39}{25}=\frac{4\sqrt{11194}i}{25} x-\frac{39}{25}=-\frac{4\sqrt{11194}i}{25}$$

\frac{39}{25} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{39+4\sqrt{11194}i}{25} x=\frac{-4\sqrt{11194}i+39}{25}$$