72x^2-72x+225=0 eching

$$72 x ^ { 2 } - 72 x + 225 = 0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{46}i}{4}+\frac{1}{2}\approx 0,5+1,695582496i$$

$$x=-\frac{\sqrt{46}i}{4}+\frac{1}{2}\approx 0,5-1,695582496i$$

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.

$$72x^{2}-72x+225=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 72 ni a, -72 ni b va 225 ni c bilan almashtiring.

$$x=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 72\times 225}}{2\times 72}$$

-72 kvadratini chiqarish.

$$x=\frac{-\left(-72\right)±\sqrt{5184-4\times 72\times 225}}{2\times 72}$$

-4 ni 72 marotabaga ko'paytirish.

$$x=\frac{-\left(-72\right)±\sqrt{5184-288\times 225}}{2\times 72}$$

-288 ni 225 marotabaga ko'paytirish.

$$x=\frac{-\left(-72\right)±\sqrt{5184-64800}}{2\times 72}$$

5184 ni -64800 ga qo'shish.

$$x=\frac{-\left(-72\right)±\sqrt{-59616}}{2\times 72}$$

-59616 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-72\right)±36\sqrt{46}i}{2\times 72}$$

-72 ning teskarisi 72 ga teng.

$$x=\frac{72±36\sqrt{46}i}{2\times 72}$$

2 ni 72 marotabaga ko'paytirish.

$$x=\frac{72±36\sqrt{46}i}{144}$$

x=\frac{72±36\sqrt{46}i}{144} tenglamasini yeching, bunda ± musbat. 72 ni 36i\sqrt{46} ga qo'shish.

$$x=\frac{72+36\sqrt{46}i}{144}$$

72+36i\sqrt{46} ni 144 ga bo'lish.

$$x=\frac{\sqrt{46}i}{4}+\frac{1}{2}$$

x=\frac{72±36\sqrt{46}i}{144} tenglamasini yeching, bunda ± manfiy. 72 dan 36i\sqrt{46} ni ayirish.

$$x=\frac{-36\sqrt{46}i+72}{144}$$

72-36i\sqrt{46} ni 144 ga bo'lish.

$$x=-\frac{\sqrt{46}i}{4}+\frac{1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{46}i}{4}+\frac{1}{2} x=-\frac{\sqrt{46}i}{4}+\frac{1}{2}$$

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.

$$72x^{2}-72x+225=0$$

Tenglamaning ikkala tarafidan 225 ni ayirish.

$$72x^{2}-72x+225-225=-225$$

O‘zidan 225 ayirilsa 0 qoladi.

$$72x^{2}-72x=-225$$

Ikki tarafini 72 ga bo‘ling.

$$\frac{72x^{2}-72x}{72}=-\frac{225}{72}$$

72 ga bo'lish 72 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{72}{72}\right)x=-\frac{225}{72}$$

-72 ni 72 ga bo'lish.

$$x^{2}-x=-\frac{225}{72}$$

\frac{-225}{72} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-x=-\frac{25}{8}$$

-1 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{2} olish uchun. Keyin, -\frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{25}{8}+\left(-\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.

$$x^{2}-x+\frac{1}{4}=-\frac{25}{8}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{8} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-x+\frac{1}{4}=-\frac{23}{8}$$

x^{2}-x+\frac{1}{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.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{23}{8}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{23}{8}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{46}i}{4} x-\frac{1}{2}=-\frac{\sqrt{46}i}{4}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{46}i}{4}+\frac{1}{2} x=-\frac{\sqrt{46}i}{4}+\frac{1}{2}$$