5x^2-48x+20=0 eching

$$5 { x }^{ 2 } -48x+20=0$$

x uchun yechish

$$x = \frac{2 \sqrt{119} + 24}{5} \approx 9,163484846$$

$$x=\frac{24-2\sqrt{119}}{5}\approx 0,436515154$$

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.

$$5x^{2}-48x+20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -48 ni b va 20 ni c bilan almashtiring.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 5\times 20}}{2\times 5}$$

-48 kvadratini chiqarish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 5\times 20}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-20\times 20}}{2\times 5}$$

-20 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-400}}{2\times 5}$$

2304 ni -400 ga qo'shish.

$$x=\frac{-\left(-48\right)±\sqrt{1904}}{2\times 5}$$

1904 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-48\right)±4\sqrt{119}}{2\times 5}$$

-48 ning teskarisi 48 ga teng.

$$x=\frac{48±4\sqrt{119}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{48±4\sqrt{119}}{10}$$

x=\frac{48±4\sqrt{119}}{10} tenglamasini yeching, bunda ± musbat. 48 ni 4\sqrt{119} ga qo'shish.

$$x=\frac{4\sqrt{119}+48}{10}$$

48+4\sqrt{119} ni 10 ga bo'lish.

$$x=\frac{2\sqrt{119}+24}{5}$$

x=\frac{48±4\sqrt{119}}{10} tenglamasini yeching, bunda ± manfiy. 48 dan 4\sqrt{119} ni ayirish.

$$x=\frac{48-4\sqrt{119}}{10}$$

48-4\sqrt{119} ni 10 ga bo'lish.

$$x=\frac{24-2\sqrt{119}}{5}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{119}+24}{5} x=\frac{24-2\sqrt{119}}{5}$$

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.

$$5x^{2}-48x+20=0$$

Tenglamaning ikkala tarafidan 20 ni ayirish.

$$5x^{2}-48x+20-20=-20$$

O‘zidan 20 ayirilsa 0 qoladi.

$$5x^{2}-48x=-20$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-48x}{5}=-\frac{20}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{48}{5}x=-\frac{20}{5}$$

-20 ni 5 ga bo'lish.

$$x^{2}-\frac{48}{5}x=-4$$

-\frac{48}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{24}{5} olish uchun. Keyin, -\frac{24}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{48}{5}x+\left(-\frac{24}{5}\right)^{2}=-4+\left(-\frac{24}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{24}{5} kvadratini chiqarish.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=-4+\frac{576}{25}$$

-4 ni \frac{576}{25} ga qo'shish.

$$x^{2}-\frac{48}{5}x+\frac{576}{25}=\frac{476}{25}$$

x^{2}-\frac{48}{5}x+\frac{576}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{24}{5}\right)^{2}=\frac{476}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{24}{5}\right)^{2}}=\sqrt{\frac{476}{25}}$$

Qisqartirish.

$$x-\frac{24}{5}=\frac{2\sqrt{119}}{5} x-\frac{24}{5}=-\frac{2\sqrt{119}}{5}$$

\frac{24}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{119}+24}{5} x=\frac{24-2\sqrt{119}}{5}$$