-4x^2+20x-47=0 eching

$$-4 { x }^{ 2 } +20x-47=0$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{22}i+5}{2}\approx 2,5-2,34520788i$$

$$x=\frac{5+\sqrt{22}i}{2}\approx 2,5+2,34520788i$$

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.

$$-4x^{2}+20x-47=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 20 ni b va -47 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-4\right)\left(-47\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+16\left(-47\right)}}{2\left(-4\right)}$$

16 ni -47 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400-752}}{2\left(-4\right)}$$

400 ni -752 ga qo'shish.

$$x=\frac{-20±\sqrt{-352}}{2\left(-4\right)}$$

-352 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±4\sqrt{22}i}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-20±4\sqrt{22}i}{-8}$$

x=\frac{-20±4\sqrt{22}i}{-8} tenglamasini yeching, bunda ± musbat. -20 ni 4i\sqrt{22} ga qo'shish.

$$x=\frac{-20+4\sqrt{22}i}{-8}$$

-20+4i\sqrt{22} ni -8 ga bo'lish.

$$x=\frac{-\sqrt{22}i+5}{2}$$

x=\frac{-20±4\sqrt{22}i}{-8} tenglamasini yeching, bunda ± manfiy. -20 dan 4i\sqrt{22} ni ayirish.

$$x=\frac{-4\sqrt{22}i-20}{-8}$$

-20-4i\sqrt{22} ni -8 ga bo'lish.

$$x=\frac{5+\sqrt{22}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{22}i+5}{2} x=\frac{5+\sqrt{22}i}{2}$$

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.

$$-4x^{2}+20x-47=0$$

47 ni tenglamaning ikkala tarafiga qo'shish.

$$-4x^{2}+20x-47-\left(-47\right)=-\left(-47\right)$$

O‘zidan -47 ayirilsa 0 qoladi.

$$-4x^{2}+20x=-\left(-47\right)$$

0 dan -47 ni ayirish.

$$-4x^{2}+20x=47$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+20x}{-4}=\frac{47}{-4}$$

-4 ga bo'lish -4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{20}{-4}x=\frac{47}{-4}$$

20 ni -4 ga bo'lish.

$$x^{2}-5x=\frac{47}{-4}$$

47 ni -4 ga bo'lish.

$$x^{2}-5x=-\frac{47}{4}$$

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

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{47}{4}+\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=\frac{-47+25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{47}{4} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-5x+\frac{25}{4}=-\frac{11}{2}$$

x^{2}-5x+\frac{25}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{11}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{11}{2}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{22}i}{2} x-\frac{5}{2}=-\frac{\sqrt{22}i}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5+\sqrt{22}i}{2} x=\frac{-\sqrt{22}i+5}{2}$$