4x+52x^2=85-25x eching

$$4x+52 { x }^{ 2 } = 85-25x$$

x uchun yechish

$$x = \frac{\sqrt{18521} - 29}{104} \approx 1,029729622$$

$$x=\frac{-\sqrt{18521}-29}{104}\approx -1,58742193$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 85 ni ayirish.

$$4x+52x^{2}-85=-25x$$

25x ni ikki tarafga qo’shing.

$$4x+52x^{2}-85+25x=0$$

29x ni olish uchun 4x va 25x ni birlashtirish.

$$29x+52x^{2}-85=0$$

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.

$$52x^{2}+29x-85=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 52 ni a, 29 ni b va -85 ni c bilan almashtiring.

$$x=\frac{-29±\sqrt{29^{2}-4\times 52\left(-85\right)}}{2\times 52}$$

29 kvadratini chiqarish.

$$x=\frac{-29±\sqrt{841-4\times 52\left(-85\right)}}{2\times 52}$$

-4 ni 52 marotabaga ko'paytirish.

$$x=\frac{-29±\sqrt{841-208\left(-85\right)}}{2\times 52}$$

-208 ni -85 marotabaga ko'paytirish.

$$x=\frac{-29±\sqrt{841+17680}}{2\times 52}$$

841 ni 17680 ga qo'shish.

$$x=\frac{-29±\sqrt{18521}}{2\times 52}$$

2 ni 52 marotabaga ko'paytirish.

$$x=\frac{-29±\sqrt{18521}}{104}$$

x=\frac{-29±\sqrt{18521}}{104} tenglamasini yeching, bunda ± musbat. -29 ni \sqrt{18521} ga qo'shish.

$$x=\frac{\sqrt{18521}-29}{104}$$

x=\frac{-29±\sqrt{18521}}{104} tenglamasini yeching, bunda ± manfiy. -29 dan \sqrt{18521} ni ayirish.

$$x=\frac{-\sqrt{18521}-29}{104}$$

Tenglama yechildi.

$$x=\frac{\sqrt{18521}-29}{104} x=\frac{-\sqrt{18521}-29}{104}$$

Kvadratni yakunlash bo'yicha qadamlar

25x ni ikki tarafga qo’shing.

$$4x+52x^{2}+25x=85$$

29x ni olish uchun 4x va 25x ni birlashtirish.

$$29x+52x^{2}=85$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$52x^{2}+29x=85$$

Ikki tarafini 52 ga bo‘ling.

$$\frac{52x^{2}+29x}{52}=\frac{85}{52}$$

52 ga bo'lish 52 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{29}{52}x=\frac{85}{52}$$

\frac{29}{52} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{29}{104} olish uchun. Keyin, \frac{29}{104} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{29}{52}x+\left(\frac{29}{104}\right)^{2}=\frac{85}{52}+\left(\frac{29}{104}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{29}{104} kvadratini chiqarish.

$$x^{2}+\frac{29}{52}x+\frac{841}{10816}=\frac{85}{52}+\frac{841}{10816}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{85}{52} ni \frac{841}{10816} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{29}{52}x+\frac{841}{10816}=\frac{18521}{10816}$$

x^{2}+\frac{29}{52}x+\frac{841}{10816} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{29}{104}\right)^{2}=\frac{18521}{10816}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{29}{104}\right)^{2}}=\sqrt{\frac{18521}{10816}}$$

Qisqartirish.

$$x+\frac{29}{104}=\frac{\sqrt{18521}}{104} x+\frac{29}{104}=-\frac{\sqrt{18521}}{104}$$

Tenglamaning ikkala tarafidan \frac{29}{104} ni ayirish.

$$x=\frac{\sqrt{18521}-29}{104} x=\frac{-\sqrt{18521}-29}{104}$$