11=1+20x-49x^2 eching

$$11=1+20x-49 { x }^{ 2 }$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{390}i+10}{49}\approx 0,204081633-0,403028932i$$

$$x=\frac{10+\sqrt{390}i}{49}\approx 0,204081633+0,403028932i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$1+20x-49x^{2}=11$$

Ikkala tarafdan 11 ni ayirish.

$$1+20x-49x^{2}-11=0$$

-10 olish uchun 1 dan 11 ni ayirish.

$$-10+20x-49x^{2}=0$$

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.

$$-49x^{2}+20x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 20 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-49\right)\left(-10\right)}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+196\left(-10\right)}}{2\left(-49\right)}$$

196 ni -10 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400-1960}}{2\left(-49\right)}$$

400 ni -1960 ga qo'shish.

$$x=\frac{-20±\sqrt{-1560}}{2\left(-49\right)}$$

-1560 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±2\sqrt{390}i}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{-20±2\sqrt{390}i}{-98}$$

x=\frac{-20±2\sqrt{390}i}{-98} tenglamasini yeching, bunda ± musbat. -20 ni 2i\sqrt{390} ga qo'shish.

$$x=\frac{-20+2\sqrt{390}i}{-98}$$

-20+2i\sqrt{390} ni -98 ga bo'lish.

$$x=\frac{-\sqrt{390}i+10}{49}$$

x=\frac{-20±2\sqrt{390}i}{-98} tenglamasini yeching, bunda ± manfiy. -20 dan 2i\sqrt{390} ni ayirish.

$$x=\frac{-2\sqrt{390}i-20}{-98}$$

-20-2i\sqrt{390} ni -98 ga bo'lish.

$$x=\frac{10+\sqrt{390}i}{49}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{390}i+10}{49} x=\frac{10+\sqrt{390}i}{49}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$1+20x-49x^{2}=11$$

Ikkala tarafdan 1 ni ayirish.

$$20x-49x^{2}=11-1$$

10 olish uchun 11 dan 1 ni ayirish.

$$20x-49x^{2}=10$$

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

$$-49x^{2}+20x=10$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}+20x}{-49}=\frac{10}{-49}$$

-49 ga bo'lish -49 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{20}{-49}x=\frac{10}{-49}$$

20 ni -49 ga bo'lish.

$$x^{2}-\frac{20}{49}x=\frac{10}{-49}$$

10 ni -49 ga bo'lish.

$$x^{2}-\frac{20}{49}x=-\frac{10}{49}$$

-\frac{20}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{10}{49} olish uchun. Keyin, -\frac{10}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{20}{49}x+\left(-\frac{10}{49}\right)^{2}=-\frac{10}{49}+\left(-\frac{10}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{10}{49} kvadratini chiqarish.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{10}{49}+\frac{100}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{10}{49} ni \frac{100}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{20}{49}x+\frac{100}{2401}=-\frac{390}{2401}$$

x^{2}-\frac{20}{49}x+\frac{100}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{10}{49}\right)^{2}=-\frac{390}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{10}{49}\right)^{2}}=\sqrt{-\frac{390}{2401}}$$

Qisqartirish.

$$x-\frac{10}{49}=\frac{\sqrt{390}i}{49} x-\frac{10}{49}=-\frac{\sqrt{390}i}{49}$$

\frac{10}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{10+\sqrt{390}i}{49} x=\frac{-\sqrt{390}i+10}{49}$$