0=4x^2-9x+14 eching

$$0 = 4 x ^ { 2 } - 9 x + 14$$

x uchun yechish (complex solution)

$$x=\frac{9+\sqrt{143}i}{8}\approx 1,125+1,494782593i$$

$$x=\frac{-\sqrt{143}i+9}{8}\approx 1,125-1,494782593i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$4x^{2}-9x+14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -9 ni b va 14 ni c bilan almashtiring.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\times 14}}{2\times 4}$$

-9 kvadratini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 14}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-16\times 14}}{2\times 4}$$

-16 ni 14 marotabaga ko'paytirish.

$$x=\frac{-\left(-9\right)±\sqrt{81-224}}{2\times 4}$$

81 ni -224 ga qo'shish.

$$x=\frac{-\left(-9\right)±\sqrt{-143}}{2\times 4}$$

-143 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-9\right)±\sqrt{143}i}{2\times 4}$$

-9 ning teskarisi 9 ga teng.

$$x=\frac{9±\sqrt{143}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{9±\sqrt{143}i}{8}$$

x=\frac{9±\sqrt{143}i}{8} tenglamasini yeching, bunda ± musbat. 9 ni i\sqrt{143} ga qo'shish.

$$x=\frac{9+\sqrt{143}i}{8}$$

x=\frac{9±\sqrt{143}i}{8} tenglamasini yeching, bunda ± manfiy. 9 dan i\sqrt{143} ni ayirish.

$$x=\frac{-\sqrt{143}i+9}{8}$$

Tenglama yechildi.

$$x=\frac{9+\sqrt{143}i}{8} x=\frac{-\sqrt{143}i+9}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$4x^{2}-9x+14=0$$

Ikkala tarafdan 14 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$4x^{2}-9x=-14$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-9x}{4}=-\frac{14}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{9}{4}x=-\frac{14}{4}$$

\frac{-14}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{9}{4}x=-\frac{7}{2}$$

-\frac{9}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{8} olish uchun. Keyin, -\frac{9}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{9}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{8} kvadratini chiqarish.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{7}{2}+\frac{81}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{2} ni \frac{81}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=-\frac{143}{64}$$

x^{2}-\frac{9}{4}x+\frac{81}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{8}\right)^{2}=-\frac{143}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{-\frac{143}{64}}$$

Qisqartirish.

$$x-\frac{9}{8}=\frac{\sqrt{143}i}{8} x-\frac{9}{8}=-\frac{\sqrt{143}i}{8}$$

\frac{9}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9+\sqrt{143}i}{8} x=\frac{-\sqrt{143}i+9}{8}$$