-144x^2+9x-9=0 eching

$$-144 { x }^{ 2 } +9x-9 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+3\sqrt{7}i}{32}\approx 0,03125+0,248039185i$$

$$x=\frac{-3\sqrt{7}i+1}{32}\approx 0,03125-0,248039185i$$

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.

$$-144x^{2}+9x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -144 ni a, 9 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-144\right)\left(-9\right)}}{2\left(-144\right)}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-144\right)\left(-9\right)}}{2\left(-144\right)}$$

-4 ni -144 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+576\left(-9\right)}}{2\left(-144\right)}$$

576 ni -9 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-5184}}{2\left(-144\right)}$$

81 ni -5184 ga qo'shish.

$$x=\frac{-9±\sqrt{-5103}}{2\left(-144\right)}$$

-5103 ning kvadrat ildizini chiqarish.

$$x=\frac{-9±27\sqrt{7}i}{2\left(-144\right)}$$

2 ni -144 marotabaga ko'paytirish.

$$x=\frac{-9±27\sqrt{7}i}{-288}$$

x=\frac{-9±27\sqrt{7}i}{-288} tenglamasini yeching, bunda ± musbat. -9 ni 27i\sqrt{7} ga qo'shish.

$$x=\frac{-9+27\sqrt{7}i}{-288}$$

-9+27i\sqrt{7} ni -288 ga bo'lish.

$$x=\frac{-3\sqrt{7}i+1}{32}$$

x=\frac{-9±27\sqrt{7}i}{-288} tenglamasini yeching, bunda ± manfiy. -9 dan 27i\sqrt{7} ni ayirish.

$$x=\frac{-27\sqrt{7}i-9}{-288}$$

-9-27i\sqrt{7} ni -288 ga bo'lish.

$$x=\frac{1+3\sqrt{7}i}{32}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{7}i+1}{32} x=\frac{1+3\sqrt{7}i}{32}$$

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.

$$-144x^{2}+9x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$-144x^{2}+9x-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$-144x^{2}+9x=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$-144x^{2}+9x=9$$

Ikki tarafini -144 ga bo‘ling.

$$\frac{-144x^{2}+9x}{-144}=\frac{9}{-144}$$

-144 ga bo'lish -144 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{9}{-144}x=\frac{9}{-144}$$

\frac{9}{-144} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{16}x=\frac{9}{-144}$$

\frac{9}{-144} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{16}x=-\frac{1}{16}$$

-\frac{1}{16} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{32} olish uchun. Keyin, -\frac{1}{32} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{16}x+\left(-\frac{1}{32}\right)^{2}=-\frac{1}{16}+\left(-\frac{1}{32}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{32} kvadratini chiqarish.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=-\frac{1}{16}+\frac{1}{1024}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{16} ni \frac{1}{1024} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{16}x+\frac{1}{1024}=-\frac{63}{1024}$$

x^{2}-\frac{1}{16}x+\frac{1}{1024} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{32}\right)^{2}=-\frac{63}{1024}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{32}\right)^{2}}=\sqrt{-\frac{63}{1024}}$$

Qisqartirish.

$$x-\frac{1}{32}=\frac{3\sqrt{7}i}{32} x-\frac{1}{32}=-\frac{3\sqrt{7}i}{32}$$

\frac{1}{32} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+3\sqrt{7}i}{32} x=\frac{-3\sqrt{7}i+1}{32}$$