(x-5)^2/2+x=3 eching

$$\frac { ( x - 5 ) ^ { 2 } } { 2 } + x = 3$$

x uchun yechish (complex solution)

$$x=4+\sqrt{3}i\approx 4+1,732050808i$$

$$x=-\sqrt{3}i+4\approx 4-1,732050808i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$\left(x-5\right)^{2}+2x=6$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-5\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-10x+25+2x=6$$

-8x ni olish uchun -10x va 2x ni birlashtirish.

$$x^{2}-8x+25=6$$

Ikkala tarafdan 6 ni ayirish.

$$x^{2}-8x+25-6=0$$

19 olish uchun 25 dan 6 ni ayirish.

$$x^{2}-8x+19=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -8 ni b va 19 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 19}}{2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 19}}{2}$$

-4 ni 19 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-76}}{2}$$

64 ni -76 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{-12}}{2}$$

-12 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±2\sqrt{3}i}{2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±2\sqrt{3}i}{2}$$

x=\frac{8±2\sqrt{3}i}{2} tenglamasini yeching, bunda ± musbat. 8 ni 2i\sqrt{3} ga qo'shish.

$$x=\frac{8+2\sqrt{3}i}{2}$$

8+2i\sqrt{3} ni 2 ga bo'lish.

$$x=4+\sqrt{3}i$$

x=\frac{8±2\sqrt{3}i}{2} tenglamasini yeching, bunda ± manfiy. 8 dan 2i\sqrt{3} ni ayirish.

$$x=\frac{-2\sqrt{3}i+8}{2}$$

8-2i\sqrt{3} ni 2 ga bo'lish.

$$x=-\sqrt{3}i+4$$

Tenglama yechildi.

$$x=4+\sqrt{3}i x=-\sqrt{3}i+4$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$\left(x-5\right)^{2}+2x=6$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-5\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-10x+25+2x=6$$

-8x ni olish uchun -10x va 2x ni birlashtirish.

$$x^{2}-8x+25=6$$

Ikkala tarafdan 25 ni ayirish.

$$x^{2}-8x=6-25$$

-19 olish uchun 6 dan 25 ni ayirish.

$$x^{2}-8x=-19$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-19+\left(-4\right)^{2}$$

-4 kvadratini chiqarish.

$$x^{2}-8x+16=-19+16$$

-19 ni 16 ga qo'shish.

$$x^{2}-8x+16=-3$$

x^{2}-8x+16 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-4\right)^{2}=-3$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-3}$$

Qisqartirish.

$$x-4=\sqrt{3}i x-4=-\sqrt{3}i$$

4 ni tenglamaning ikkala tarafiga qo'shish.

$$x=4+\sqrt{3}i x=-\sqrt{3}i+4$$