-3x^2+16x+128=0 eching

$$-3 { x }^{ 2 } +16x+128=0$$

x uchun yechish

$$x = \frac{8 \sqrt{7} + 8}{3} \approx 9,722003496$$

$$x=\frac{8-8\sqrt{7}}{3}\approx -4,388670163$$

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.

$$-3x^{2}+16x+128=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 16 ni b va 128 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-3\right)\times 128}}{2\left(-3\right)}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\left(-3\right)\times 128}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+12\times 128}}{2\left(-3\right)}$$

12 ni 128 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+1536}}{2\left(-3\right)}$$

256 ni 1536 ga qo'shish.

$$x=\frac{-16±\sqrt{1792}}{2\left(-3\right)}$$

1792 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±16\sqrt{7}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-16±16\sqrt{7}}{-6}$$

x=\frac{-16±16\sqrt{7}}{-6} tenglamasini yeching, bunda ± musbat. -16 ni 16\sqrt{7} ga qo'shish.

$$x=\frac{16\sqrt{7}-16}{-6}$$

-16+16\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{8-8\sqrt{7}}{3}$$

x=\frac{-16±16\sqrt{7}}{-6} tenglamasini yeching, bunda ± manfiy. -16 dan 16\sqrt{7} ni ayirish.

$$x=\frac{-16\sqrt{7}-16}{-6}$$

-16-16\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{8\sqrt{7}+8}{3}$$

Tenglama yechildi.

$$x=\frac{8-8\sqrt{7}}{3} x=\frac{8\sqrt{7}+8}{3}$$

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.

$$-3x^{2}+16x+128=0$$

Tenglamaning ikkala tarafidan 128 ni ayirish.

$$-3x^{2}+16x+128-128=-128$$

O‘zidan 128 ayirilsa 0 qoladi.

$$-3x^{2}+16x=-128$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+16x}{-3}=-\frac{128}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{16}{-3}x=-\frac{128}{-3}$$

16 ni -3 ga bo'lish.

$$x^{2}-\frac{16}{3}x=-\frac{128}{-3}$$

-128 ni -3 ga bo'lish.

$$x^{2}-\frac{16}{3}x=\frac{128}{3}$$

-\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{8}{3} olish uchun. Keyin, -\frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{16}{3}x+\left(-\frac{8}{3}\right)^{2}=\frac{128}{3}+\left(-\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{8}{3} kvadratini chiqarish.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{128}{3}+\frac{64}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{128}{3} ni \frac{64}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{448}{9}$$

x^{2}-\frac{16}{3}x+\frac{64}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{8}{3}\right)^{2}=\frac{448}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{8}{3}\right)^{2}}=\sqrt{\frac{448}{9}}$$

Qisqartirish.

$$x-\frac{8}{3}=\frac{8\sqrt{7}}{3} x-\frac{8}{3}=-\frac{8\sqrt{7}}{3}$$

\frac{8}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{8\sqrt{7}+8}{3} x=\frac{8-8\sqrt{7}}{3}$$