(100+10x)(60-x)=2240 eching

$$( 100 + 10 x ) ( 60 - x ) = 2240$$

x uchun yechish

$$x=\sqrt{1001}+25\approx 56,638584039$$

$$x=25-\sqrt{1001}\approx -6,638584039$$

Kvadrat tenglamalaridan foydalanish qoidalari

100+10x ga 60-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6000+500x-10x^{2}=2240$$

Ikkala tarafdan 2240 ni ayirish.

$$6000+500x-10x^{2}-2240=0$$

3760 olish uchun 6000 dan 2240 ni ayirish.

$$3760+500x-10x^{2}=0$$

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.

$$-10x^{2}+500x+3760=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -10 ni a, 500 ni b va 3760 ni c bilan almashtiring.

$$x=\frac{-500±\sqrt{500^{2}-4\left(-10\right)\times 3760}}{2\left(-10\right)}$$

500 kvadratini chiqarish.

$$x=\frac{-500±\sqrt{250000-4\left(-10\right)\times 3760}}{2\left(-10\right)}$$

-4 ni -10 marotabaga ko'paytirish.

$$x=\frac{-500±\sqrt{250000+40\times 3760}}{2\left(-10\right)}$$

40 ni 3760 marotabaga ko'paytirish.

$$x=\frac{-500±\sqrt{250000+150400}}{2\left(-10\right)}$$

250000 ni 150400 ga qo'shish.

$$x=\frac{-500±\sqrt{400400}}{2\left(-10\right)}$$

400400 ning kvadrat ildizini chiqarish.

$$x=\frac{-500±20\sqrt{1001}}{2\left(-10\right)}$$

2 ni -10 marotabaga ko'paytirish.

$$x=\frac{-500±20\sqrt{1001}}{-20}$$

x=\frac{-500±20\sqrt{1001}}{-20} tenglamasini yeching, bunda ± musbat. -500 ni 20\sqrt{1001} ga qo'shish.

$$x=\frac{20\sqrt{1001}-500}{-20}$$

-500+20\sqrt{1001} ni -20 ga bo'lish.

$$x=25-\sqrt{1001}$$

x=\frac{-500±20\sqrt{1001}}{-20} tenglamasini yeching, bunda ± manfiy. -500 dan 20\sqrt{1001} ni ayirish.

$$x=\frac{-20\sqrt{1001}-500}{-20}$$

-500-20\sqrt{1001} ni -20 ga bo'lish.

$$x=\sqrt{1001}+25$$

Tenglama yechildi.

$$x=25-\sqrt{1001} x=\sqrt{1001}+25$$

Kvadratni yakunlash bo'yicha qadamlar

100+10x ga 60-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6000+500x-10x^{2}=2240$$

Ikkala tarafdan 6000 ni ayirish.

$$500x-10x^{2}=2240-6000$$

-3760 olish uchun 2240 dan 6000 ni ayirish.

$$500x-10x^{2}=-3760$$

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

$$-10x^{2}+500x=-3760$$

Ikki tarafini -10 ga bo‘ling.

$$\frac{-10x^{2}+500x}{-10}=-\frac{3760}{-10}$$

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

$$x^{2}+\frac{500}{-10}x=-\frac{3760}{-10}$$

500 ni -10 ga bo'lish.

$$x^{2}-50x=-\frac{3760}{-10}$$

-3760 ni -10 ga bo'lish.

$$x^{2}-50x=376$$

-50 ni bo‘lish, x shartining koeffitsienti, 2 ga -25 olish uchun. Keyin, -25 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-50x+\left(-25\right)^{2}=376+\left(-25\right)^{2}$$

-25 kvadratini chiqarish.

$$x^{2}-50x+625=376+625$$

376 ni 625 ga qo'shish.

$$x^{2}-50x+625=1001$$

x^{2}-50x+625 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-25\right)^{2}=1001$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{1001}$$

Qisqartirish.

$$x-25=\sqrt{1001} x-25=-\sqrt{1001}$$

25 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{1001}+25 x=25-\sqrt{1001}$$