4x^2-11x+30=16 eching

$$4 x ^ { 2 } - 11 x + 30 = 16$$

x uchun yechish (complex solution)

$$x=\frac{11+\sqrt{103}i}{8}\approx 1,375+1,268611446i$$

$$x=\frac{-\sqrt{103}i+11}{8}\approx 1,375-1,268611446i$$

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.

$$4x^{2}-11x+30=16$$

Tenglamaning ikkala tarafidan 16 ni ayirish.

$$4x^{2}-11x+30-16=16-16$$

O‘zidan 16 ayirilsa 0 qoladi.

$$4x^{2}-11x+30-16=0$$

30 dan 16 ni ayirish.

$$4x^{2}-11x+14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -11 ni b va 14 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 14}}{2\times 4}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 14}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\times 14}}{2\times 4}$$

-16 ni 14 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-224}}{2\times 4}$$

121 ni -224 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{-103}}{2\times 4}$$

-103 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{103}i}{2\times 4}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{103}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{11±\sqrt{103}i}{8}$$

x=\frac{11±\sqrt{103}i}{8} tenglamasini yeching, bunda ± musbat. 11 ni i\sqrt{103} ga qo'shish.

$$x=\frac{11+\sqrt{103}i}{8}$$

x=\frac{11±\sqrt{103}i}{8} tenglamasini yeching, bunda ± manfiy. 11 dan i\sqrt{103} ni ayirish.

$$x=\frac{-\sqrt{103}i+11}{8}$$

Tenglama yechildi.

$$x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}$$

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.

$$4x^{2}-11x+30=16$$

Tenglamaning ikkala tarafidan 30 ni ayirish.

$$4x^{2}-11x+30-30=16-30$$

O‘zidan 30 ayirilsa 0 qoladi.

$$4x^{2}-11x=16-30$$

16 dan 30 ni ayirish.

$$4x^{2}-11x=-14$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-11x}{4}=-\frac{14}{4}$$

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

$$x^{2}-\frac{11}{4}x=-\frac{14}{4}$$

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

$$x^{2}-\frac{11}{4}x=-\frac{7}{2}$$

-\frac{11}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{8} olish uchun. Keyin, -\frac{11}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{11}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{8} kvadratini chiqarish.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{7}{2}+\frac{121}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{2} ni \frac{121}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{103}{64}$$

x^{2}-\frac{11}{4}x+\frac{121}{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.

$$\left(x-\frac{11}{8}\right)^{2}=-\frac{103}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{-\frac{103}{64}}$$

Qisqartirish.

$$x-\frac{11}{8}=\frac{\sqrt{103}i}{8} x-\frac{11}{8}=-\frac{\sqrt{103}i}{8}$$

\frac{11}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}$$