(12+x)*(1*2-x)=108 eching

$$( 12 + x ) \times ( 1 \cdot 2 - x ) = 108$$

x uchun yechish (complex solution)

$$x=-\sqrt{59}i-5\approx -5-7,681145748i$$

$$x=-5+\sqrt{59}i\approx -5+7,681145748i$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 hosil qilish uchun 1 va 2 ni ko'paytirish.

$$\left(12+x\right)\left(2-x\right)=108$$

12+x ga 2-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$24-10x-x^{2}=108$$

Ikkala tarafdan 108 ni ayirish.

$$24-10x-x^{2}-108=0$$

-84 olish uchun 24 dan 108 ni ayirish.

$$-84-10x-x^{2}=0$$

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.

$$-x^{2}-10x-84=0$$

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 -84 ni c bilan almashtiring.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

-10 kvadratini chiqarish.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\left(-1\right)\left(-84\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100+4\left(-84\right)}}{2\left(-1\right)}$$

4 ni -84 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-336}}{2\left(-1\right)}$$

100 ni -336 ga qo'shish.

$$x=\frac{-\left(-10\right)±\sqrt{-236}}{2\left(-1\right)}$$

-236 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-10\right)±2\sqrt{59}i}{2\left(-1\right)}$$

-10 ning teskarisi 10 ga teng.

$$x=\frac{10±2\sqrt{59}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{10±2\sqrt{59}i}{-2}$$

x=\frac{10±2\sqrt{59}i}{-2} tenglamasini yeching, bunda ± musbat. 10 ni 2i\sqrt{59} ga qo'shish.

$$x=\frac{10+2\sqrt{59}i}{-2}$$

10+2i\sqrt{59} ni -2 ga bo'lish.

$$x=-\sqrt{59}i-5$$

x=\frac{10±2\sqrt{59}i}{-2} tenglamasini yeching, bunda ± manfiy. 10 dan 2i\sqrt{59} ni ayirish.

$$x=\frac{-2\sqrt{59}i+10}{-2}$$

10-2i\sqrt{59} ni -2 ga bo'lish.

$$x=-5+\sqrt{59}i$$

Tenglama yechildi.

$$x=-\sqrt{59}i-5 x=-5+\sqrt{59}i$$

Kvadratni yakunlash bo'yicha qadamlar

2 hosil qilish uchun 1 va 2 ni ko'paytirish.

$$\left(12+x\right)\left(2-x\right)=108$$

12+x ga 2-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$24-10x-x^{2}=108$$

Ikkala tarafdan 24 ni ayirish.

$$-10x-x^{2}=108-24$$

84 olish uchun 108 dan 24 ni ayirish.

$$-10x-x^{2}=84$$

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

$$-x^{2}-10x=84$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-10x}{-1}=\frac{84}{-1}$$

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

$$x^{2}+\left(-\frac{10}{-1}\right)x=\frac{84}{-1}$$

-10 ni -1 ga bo'lish.

$$x^{2}+10x=\frac{84}{-1}$$

84 ni -1 ga bo'lish.

$$x^{2}+10x=-84$$

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.

$$x^{2}+10x+5^{2}=-84+5^{2}$$

5 kvadratini chiqarish.

$$x^{2}+10x+25=-84+25$$

-84 ni 25 ga qo'shish.

$$x^{2}+10x+25=-59$$

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.

$$\left(x+5\right)^{2}=-59$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{-59}$$

Qisqartirish.

$$x+5=\sqrt{59}i x+5=-\sqrt{59}i$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$x=-5+\sqrt{59}i x=-\sqrt{59}i-5$$