f^2-3f=-5 eching

$$f ^ { 2 } - 3 f = - 5$$

f uchun yechish

$$f=\frac{3+\sqrt{11}i}{2}\approx 1,5+1,658312395i$$

$$f=\frac{-\sqrt{11}i+3}{2}\approx 1,5-1,658312395i$$

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.

$$f^{2}-3f=-5$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$f^{2}-3f-\left(-5\right)=-5-\left(-5\right)$$

O‘zidan -5 ayirilsa 0 qoladi.

$$f^{2}-3f-\left(-5\right)=0$$

0 dan -5 ni ayirish.

$$f^{2}-3f+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -3 ni b va 5 ni c bilan almashtiring.

$$f=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 5}}{2}$$

-3 kvadratini chiqarish.

$$f=\frac{-\left(-3\right)±\sqrt{9-4\times 5}}{2}$$

-4 ni 5 marotabaga ko'paytirish.

$$f=\frac{-\left(-3\right)±\sqrt{9-20}}{2}$$

9 ni -20 ga qo'shish.

$$f=\frac{-\left(-3\right)±\sqrt{-11}}{2}$$

-11 ning kvadrat ildizini chiqarish.

$$f=\frac{-\left(-3\right)±\sqrt{11}i}{2}$$

-3 ning teskarisi 3 ga teng.

$$f=\frac{3±\sqrt{11}i}{2}$$

f=\frac{3±\sqrt{11}i}{2} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{11} ga qo'shish.

$$f=\frac{3+\sqrt{11}i}{2}$$

f=\frac{3±\sqrt{11}i}{2} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{11} ni ayirish.

$$f=\frac{-\sqrt{11}i+3}{2}$$

Tenglama yechildi.

$$f=\frac{3+\sqrt{11}i}{2} f=\frac{-\sqrt{11}i+3}{2}$$

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.

$$f^{2}-3f=-5$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$f^{2}-3f+\left(-\frac{3}{2}\right)^{2}=-5+\left(-\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$f^{2}-3f+\frac{9}{4}=-5+\frac{9}{4}$$

-5 ni \frac{9}{4} ga qo'shish.

$$f^{2}-3f+\frac{9}{4}=-\frac{11}{4}$$

f^{2}-3f+\frac{9}{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.

$$\left(f-\frac{3}{2}\right)^{2}=-\frac{11}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(f-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Qisqartirish.

$$f-\frac{3}{2}=\frac{\sqrt{11}i}{2} f-\frac{3}{2}=-\frac{\sqrt{11}i}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$f=\frac{3+\sqrt{11}i}{2} f=\frac{-\sqrt{11}i+3}{2}$$