2x^2-9x+18=0 eching

$$2 x ^ { 2 } - 9 x + 18 = 0$$

x uchun yechish (complex solution)

$$x=\frac{9+3\sqrt{7}i}{4}\approx 2,25+1,984313483i$$

$$x=\frac{-3\sqrt{7}i+9}{4}\approx 2,25-1,984313483i$$

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.

$$2x^{2}-9x+18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -9 ni b va 18 ni c bilan almashtiring.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\times 18}}{2\times 2}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\times 18}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\times 18}}{2\times 2}$$

-8 ni 18 marotabaga ko'paytirish.

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

81 ni -144 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{-63}}{2\times 2}$$

-63 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-9\right)±3\sqrt{7}i}{2\times 2}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±3\sqrt{7}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{9±3\sqrt{7}i}{4}$$

x=\frac{9±3\sqrt{7}i}{4} tenglamasini yeching, bunda ± musbat. 9 ni 3i\sqrt{7} ga qo'shish.

$$x=\frac{9+3\sqrt{7}i}{4}$$

x=\frac{9±3\sqrt{7}i}{4} tenglamasini yeching, bunda ± manfiy. 9 dan 3i\sqrt{7} ni ayirish.

$$x=\frac{-3\sqrt{7}i+9}{4}$$

Tenglama yechildi.

$$x=\frac{9+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+9}{4}$$

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.

$$2x^{2}-9x+18=0$$

Tenglamaning ikkala tarafidan 18 ni ayirish.

$$2x^{2}-9x+18-18=-18$$

O‘zidan 18 ayirilsa 0 qoladi.

$$2x^{2}-9x=-18$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-9x}{2}=-\frac{18}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{9}{2}x=-\frac{18}{2}$$

-18 ni 2 ga bo'lish.

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

-\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{4} olish uchun. Keyin, -\frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{4} kvadratini chiqarish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-9+\frac{81}{16}$$

-9 ni \frac{81}{16} ga qo'shish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{63}{16}$$

x^{2}-\frac{9}{2}x+\frac{81}{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.

$$\left(x-\frac{9}{4}\right)^{2}=-\frac{63}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{-\frac{63}{16}}$$

Qisqartirish.

$$x-\frac{9}{4}=\frac{3\sqrt{7}i}{4} x-\frac{9}{4}=-\frac{3\sqrt{7}i}{4}$$

\frac{9}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+9}{4}$$