24a^2-60a+352=0 eching

$$24 a ^ { 2 } - 60 a + 352 = 0$$

a uchun yechish

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4}\approx 1,25+3,619967771i$$

$$a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}\approx 1,25-3,619967771i$$

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.

$$24a^{2}-60a+352=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 24 ni a, -60 ni b va 352 ni c bilan almashtiring.

$$a=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 24\times 352}}{2\times 24}$$

-60 kvadratini chiqarish.

$$a=\frac{-\left(-60\right)±\sqrt{3600-4\times 24\times 352}}{2\times 24}$$

-4 ni 24 marotabaga ko'paytirish.

$$a=\frac{-\left(-60\right)±\sqrt{3600-96\times 352}}{2\times 24}$$

-96 ni 352 marotabaga ko'paytirish.

$$a=\frac{-\left(-60\right)±\sqrt{3600-33792}}{2\times 24}$$

3600 ni -33792 ga qo'shish.

$$a=\frac{-\left(-60\right)±\sqrt{-30192}}{2\times 24}$$

-30192 ning kvadrat ildizini chiqarish.

$$a=\frac{-\left(-60\right)±4\sqrt{1887}i}{2\times 24}$$

-60 ning teskarisi 60 ga teng.

$$a=\frac{60±4\sqrt{1887}i}{2\times 24}$$

2 ni 24 marotabaga ko'paytirish.

$$a=\frac{60±4\sqrt{1887}i}{48}$$

a=\frac{60±4\sqrt{1887}i}{48} tenglamasini yeching, bunda ± musbat. 60 ni 4i\sqrt{1887} ga qo'shish.

$$a=\frac{60+4\sqrt{1887}i}{48}$$

60+4i\sqrt{1887} ni 48 ga bo'lish.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

a=\frac{60±4\sqrt{1887}i}{48} tenglamasini yeching, bunda ± manfiy. 60 dan 4i\sqrt{1887} ni ayirish.

$$a=\frac{-4\sqrt{1887}i+60}{48}$$

60-4i\sqrt{1887} ni 48 ga bo'lish.

$$a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

Tenglama yechildi.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4} a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$

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.

$$24a^{2}-60a+352=0$$

Tenglamaning ikkala tarafidan 352 ni ayirish.

$$24a^{2}-60a+352-352=-352$$

O‘zidan 352 ayirilsa 0 qoladi.

$$24a^{2}-60a=-352$$

Ikki tarafini 24 ga bo‘ling.

$$\frac{24a^{2}-60a}{24}=-\frac{352}{24}$$

24 ga bo'lish 24 ga ko'paytirishni bekor qiladi.

$$a^{2}+\left(-\frac{60}{24}\right)a=-\frac{352}{24}$$

\frac{-60}{24} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a^{2}-\frac{5}{2}a=-\frac{352}{24}$$

\frac{-352}{24} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a^{2}-\frac{5}{2}a=-\frac{44}{3}$$

-\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{4} olish uchun. Keyin, -\frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}-\frac{5}{2}a+\left(-\frac{5}{4}\right)^{2}=-\frac{44}{3}+\left(-\frac{5}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{4} kvadratini chiqarish.

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=-\frac{44}{3}+\frac{25}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{44}{3} ni \frac{25}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$a^{2}-\frac{5}{2}a+\frac{25}{16}=-\frac{629}{48}$$

a^{2}-\frac{5}{2}a+\frac{25}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(a-\frac{5}{4}\right)^{2}=-\frac{629}{48}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{629}{48}}$$

Qisqartirish.

$$a-\frac{5}{4}=\frac{\sqrt{1887}i}{12} a-\frac{5}{4}=-\frac{\sqrt{1887}i}{12}$$

\frac{5}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$a=\frac{\sqrt{1887}i}{12}+\frac{5}{4} a=-\frac{\sqrt{1887}i}{12}+\frac{5}{4}$$