(9-x)(x-4)=3x eching

$$( 9 - x ) ( x - 4 ) = 3 x$$

x uchun yechish (complex solution)

$$x=-\sqrt{11}i+5\approx 5-3,31662479i$$

$$x=5+\sqrt{11}i\approx 5+3,31662479i$$

Kvadrat tenglamalaridan foydalanish qoidalari

9-x ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$13x-36-x^{2}=3x$$

Ikkala tarafdan 3x ni ayirish.

$$13x-36-x^{2}-3x=0$$

10x ni olish uchun 13x va -3x ni birlashtirish.

$$10x-36-x^{2}=0$$

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.

$$-x^{2}+10x-36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 10 ni b va -36 ni c bilan almashtiring.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

10 kvadratini chiqarish.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\left(-36\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100+4\left(-36\right)}}{2\left(-1\right)}$$

4 ni -36 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100-144}}{2\left(-1\right)}$$

100 ni -144 ga qo'shish.

$$x=\frac{-10±\sqrt{-44}}{2\left(-1\right)}$$

-44 ning kvadrat ildizini chiqarish.

$$x=\frac{-10±2\sqrt{11}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-10±2\sqrt{11}i}{-2}$$

x=\frac{-10±2\sqrt{11}i}{-2} tenglamasini yeching, bunda ± musbat. -10 ni 2i\sqrt{11} ga qo'shish.

$$x=\frac{-10+2\sqrt{11}i}{-2}$$

-10+2i\sqrt{11} ni -2 ga bo'lish.

$$x=-\sqrt{11}i+5$$

x=\frac{-10±2\sqrt{11}i}{-2} tenglamasini yeching, bunda ± manfiy. -10 dan 2i\sqrt{11} ni ayirish.

$$x=\frac{-2\sqrt{11}i-10}{-2}$$

-10-2i\sqrt{11} ni -2 ga bo'lish.

$$x=5+\sqrt{11}i$$

Tenglama yechildi.

$$x=-\sqrt{11}i+5 x=5+\sqrt{11}i$$

Kvadratni yakunlash bo'yicha qadamlar

9-x ga x-4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$13x-36-x^{2}=3x$$

Ikkala tarafdan 3x ni ayirish.

$$13x-36-x^{2}-3x=0$$

10x ni olish uchun 13x va -3x ni birlashtirish.

$$10x-36-x^{2}=0$$

36 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$10x-x^{2}=36$$

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

$$-x^{2}+10x=36$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+10x}{-1}=\frac{36}{-1}$$

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

$$x^{2}+\frac{10}{-1}x=\frac{36}{-1}$$

10 ni -1 ga bo'lish.

$$x^{2}-10x=\frac{36}{-1}$$

36 ni -1 ga bo'lish.

$$x^{2}-10x=-36$$

-10 ni bo‘lish, x shartining koeffitsienti, 2 ga -5 olish uchun. Keyin, -5 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-10x+\left(-5\right)^{2}=-36+\left(-5\right)^{2}$$

-5 kvadratini chiqarish.

$$x^{2}-10x+25=-36+25$$

-36 ni 25 ga qo'shish.

$$x^{2}-10x+25=-11$$

x^{2}-10x+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.

$$\left(x-5\right)^{2}=-11$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-11}$$

Qisqartirish.

$$x-5=\sqrt{11}i x-5=-\sqrt{11}i$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$x=5+\sqrt{11}i x=-\sqrt{11}i+5$$