(20-x)left(100+10xright)=2240 eching

$$(20-x) \left( 100+10x \right) = 2240$$

x uchun yechish

$$x=4$$

$$x=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

20-x ga 100+10x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+100x-10x^{2}=2240$$

Ikkala tarafdan 2240 ni ayirish.

$$2000+100x-10x^{2}-2240=0$$

-240 olish uchun 2000 dan 2240 ni ayirish.

$$-240+100x-10x^{2}=0$$

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.

$$-10x^{2}+100x-240=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -10 ni a, 100 ni b va -240 ni c bilan almashtiring.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

100 kvadratini chiqarish.

$$x=\frac{-100±\sqrt{10000-4\left(-10\right)\left(-240\right)}}{2\left(-10\right)}$$

-4 ni -10 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000+40\left(-240\right)}}{2\left(-10\right)}$$

40 ni -240 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000-9600}}{2\left(-10\right)}$$

10000 ni -9600 ga qo'shish.

$$x=\frac{-100±\sqrt{400}}{2\left(-10\right)}$$

400 ning kvadrat ildizini chiqarish.

$$x=\frac{-100±20}{2\left(-10\right)}$$

2 ni -10 marotabaga ko'paytirish.

$$x=\frac{-100±20}{-20}$$

x=\frac{-100±20}{-20} tenglamasini yeching, bunda ± musbat. -100 ni 20 ga qo'shish.

$$x=-\frac{80}{-20}$$

-80 ni -20 ga bo'lish.

$$x=4$$

x=\frac{-100±20}{-20} tenglamasini yeching, bunda ± manfiy. -100 dan 20 ni ayirish.

$$x=-\frac{120}{-20}$$

-120 ni -20 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=4 x=6$$

Kvadratni yakunlash bo'yicha qadamlar

20-x ga 100+10x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+100x-10x^{2}=2240$$

Ikkala tarafdan 2000 ni ayirish.

$$100x-10x^{2}=2240-2000$$

240 olish uchun 2240 dan 2000 ni ayirish.

$$100x-10x^{2}=240$$

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

$$-10x^{2}+100x=240$$

Ikki tarafini -10 ga bo‘ling.

$$\frac{-10x^{2}+100x}{-10}=\frac{240}{-10}$$

-10 ga bo'lish -10 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{100}{-10}x=\frac{240}{-10}$$

100 ni -10 ga bo'lish.

$$x^{2}-10x=\frac{240}{-10}$$

240 ni -10 ga bo'lish.

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

-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.

$$x^{2}-10x+\left(-5\right)^{2}=-24+\left(-5\right)^{2}$$

-5 kvadratini chiqarish.

$$x^{2}-10x+25=-24+25$$

-24 ni 25 ga qo'shish.

$$x^{2}-10x+25=1$$

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.

$$\left(x-5\right)^{2}=1$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{1}$$

Qisqartirish.

$$x-5=1 x-5=-1$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$x=6 x=4$$