2(x-6)+37=41+x^2 eching

$$2(x-6)+37=41+ { x }^{ 2 }$$

x uchun yechish (complex solution)

$$x=-\sqrt{15}i+1\approx 1-3,872983346i$$

$$x=1+\sqrt{15}i\approx 1+3,872983346i$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 ga x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x-12+37=41+x^{2}$$

25 olish uchun -12 va 37'ni qo'shing.

$$2x+25=41+x^{2}$$

Ikkala tarafdan 41 ni ayirish.

$$2x+25-41=x^{2}$$

-16 olish uchun 25 dan 41 ni ayirish.

$$2x-16=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x-16-x^{2}=0$$

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.

$$-x^{2}+2x-16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 2 ni b va -16 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+4\left(-16\right)}}{2\left(-1\right)}$$

4 ni -16 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-64}}{2\left(-1\right)}$$

4 ni -64 ga qo'shish.

$$x=\frac{-2±\sqrt{-60}}{2\left(-1\right)}$$

-60 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{15}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{15}i}{-2}$$

x=\frac{-2±2\sqrt{15}i}{-2} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{15} ga qo'shish.

$$x=\frac{-2+2\sqrt{15}i}{-2}$$

-2+2i\sqrt{15} ni -2 ga bo'lish.

$$x=-\sqrt{15}i+1$$

x=\frac{-2±2\sqrt{15}i}{-2} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{15} ni ayirish.

$$x=\frac{-2\sqrt{15}i-2}{-2}$$

-2-2i\sqrt{15} ni -2 ga bo'lish.

$$x=1+\sqrt{15}i$$

Tenglama yechildi.

$$x=-\sqrt{15}i+1 x=1+\sqrt{15}i$$

Kvadratni yakunlash bo'yicha qadamlar

2 ga x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x-12+37=41+x^{2}$$

25 olish uchun -12 va 37'ni qo'shing.

$$2x+25=41+x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$2x+25-x^{2}=41$$

Ikkala tarafdan 25 ni ayirish.

$$2x-x^{2}=41-25$$

16 olish uchun 41 dan 25 ni ayirish.

$$2x-x^{2}=16$$

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

$$-x^{2}+2x=16$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+2x}{-1}=\frac{16}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{-1}x=\frac{16}{-1}$$

2 ni -1 ga bo'lish.

$$x^{2}-2x=\frac{16}{-1}$$

16 ni -1 ga bo'lish.

$$x^{2}-2x=-16$$

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=-16+1$$

-16 ni 1 ga qo'shish.

$$x^{2}-2x+1=-15$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=-15$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-15}$$

Qisqartirish.

$$x-1=\sqrt{15}i x-1=-\sqrt{15}i$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=1+\sqrt{15}i x=-\sqrt{15}i+1$$