27=22t-5t^2 eching

$$27 = 22 t - 5 t ^ { 2 }$$

t uchun yechish

$$t=\frac{11+\sqrt{14}i}{5}\approx 2,2+0,748331477i$$

$$t=\frac{-\sqrt{14}i+11}{5}\approx 2,2-0,748331477i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$22t-5t^{2}=27$$

Ikkala tarafdan 27 ni ayirish.

$$22t-5t^{2}-27=0$$

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.

$$-5t^{2}+22t-27=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 22 ni b va -27 ni c bilan almashtiring.

$$t=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-27\right)}}{2\left(-5\right)}$$

22 kvadratini chiqarish.

$$t=\frac{-22±\sqrt{484-4\left(-5\right)\left(-27\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$t=\frac{-22±\sqrt{484+20\left(-27\right)}}{2\left(-5\right)}$$

20 ni -27 marotabaga ko'paytirish.

$$t=\frac{-22±\sqrt{484-540}}{2\left(-5\right)}$$

484 ni -540 ga qo'shish.

$$t=\frac{-22±\sqrt{-56}}{2\left(-5\right)}$$

-56 ning kvadrat ildizini chiqarish.

$$t=\frac{-22±2\sqrt{14}i}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$t=\frac{-22±2\sqrt{14}i}{-10}$$

t=\frac{-22±2\sqrt{14}i}{-10} tenglamasini yeching, bunda ± musbat. -22 ni 2i\sqrt{14} ga qo'shish.

$$t=\frac{-22+2\sqrt{14}i}{-10}$$

-22+2i\sqrt{14} ni -10 ga bo'lish.

$$t=\frac{-\sqrt{14}i+11}{5}$$

t=\frac{-22±2\sqrt{14}i}{-10} tenglamasini yeching, bunda ± manfiy. -22 dan 2i\sqrt{14} ni ayirish.

$$t=\frac{-2\sqrt{14}i-22}{-10}$$

-22-2i\sqrt{14} ni -10 ga bo'lish.

$$t=\frac{11+\sqrt{14}i}{5}$$

Tenglama yechildi.

$$t=\frac{-\sqrt{14}i+11}{5} t=\frac{11+\sqrt{14}i}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$22t-5t^{2}=27$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-5t^{2}+22t=27$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5t^{2}+22t}{-5}=\frac{27}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$t^{2}+\frac{22}{-5}t=\frac{27}{-5}$$

22 ni -5 ga bo'lish.

$$t^{2}-\frac{22}{5}t=\frac{27}{-5}$$

27 ni -5 ga bo'lish.

$$t^{2}-\frac{22}{5}t=-\frac{27}{5}$$

-\frac{22}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{5} olish uchun. Keyin, -\frac{11}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{22}{5}t+\left(-\frac{11}{5}\right)^{2}=-\frac{27}{5}+\left(-\frac{11}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{5} kvadratini chiqarish.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=-\frac{27}{5}+\frac{121}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{27}{5} ni \frac{121}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=-\frac{14}{25}$$

t^{2}-\frac{22}{5}t+\frac{121}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{11}{5}\right)^{2}=-\frac{14}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{11}{5}\right)^{2}}=\sqrt{-\frac{14}{25}}$$

Qisqartirish.

$$t-\frac{11}{5}=\frac{\sqrt{14}i}{5} t-\frac{11}{5}=-\frac{\sqrt{14}i}{5}$$

\frac{11}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{11+\sqrt{14}i}{5} t=\frac{-\sqrt{14}i+11}{5}$$