(10-x)(200+50x)=1250 eching

$$(10-x)(200+50x)=1250$$

x uchun yechish

$$x=2\sqrt{6}+3\approx 7,898979486$$

$$x=3-2\sqrt{6}\approx -1,898979486$$

Kvadrat tenglamalaridan foydalanish qoidalari

10-x ga 200+50x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+300x-50x^{2}=1250$$

Ikkala tarafdan 1250 ni ayirish.

$$2000+300x-50x^{2}-1250=0$$

750 olish uchun 2000 dan 1250 ni ayirish.

$$750+300x-50x^{2}=0$$

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.

$$-50x^{2}+300x+750=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -50 ni a, 300 ni b va 750 ni c bilan almashtiring.

$$x=\frac{-300±\sqrt{300^{2}-4\left(-50\right)\times 750}}{2\left(-50\right)}$$

300 kvadratini chiqarish.

$$x=\frac{-300±\sqrt{90000-4\left(-50\right)\times 750}}{2\left(-50\right)}$$

-4 ni -50 marotabaga ko'paytirish.

$$x=\frac{-300±\sqrt{90000+200\times 750}}{2\left(-50\right)}$$

200 ni 750 marotabaga ko'paytirish.

$$x=\frac{-300±\sqrt{90000+150000}}{2\left(-50\right)}$$

90000 ni 150000 ga qo'shish.

$$x=\frac{-300±\sqrt{240000}}{2\left(-50\right)}$$

240000 ning kvadrat ildizini chiqarish.

$$x=\frac{-300±200\sqrt{6}}{2\left(-50\right)}$$

2 ni -50 marotabaga ko'paytirish.

$$x=\frac{-300±200\sqrt{6}}{-100}$$

x=\frac{-300±200\sqrt{6}}{-100} tenglamasini yeching, bunda ± musbat. -300 ni 200\sqrt{6} ga qo'shish.

$$x=\frac{200\sqrt{6}-300}{-100}$$

-300+200\sqrt{6} ni -100 ga bo'lish.

$$x=3-2\sqrt{6}$$

x=\frac{-300±200\sqrt{6}}{-100} tenglamasini yeching, bunda ± manfiy. -300 dan 200\sqrt{6} ni ayirish.

$$x=\frac{-200\sqrt{6}-300}{-100}$$

-300-200\sqrt{6} ni -100 ga bo'lish.

$$x=2\sqrt{6}+3$$

Tenglama yechildi.

$$x=3-2\sqrt{6} x=2\sqrt{6}+3$$

Kvadratni yakunlash bo'yicha qadamlar

10-x ga 200+50x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2000+300x-50x^{2}=1250$$

Ikkala tarafdan 2000 ni ayirish.

$$300x-50x^{2}=1250-2000$$

-750 olish uchun 1250 dan 2000 ni ayirish.

$$300x-50x^{2}=-750$$

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

$$-50x^{2}+300x=-750$$

Ikki tarafini -50 ga bo‘ling.

$$\frac{-50x^{2}+300x}{-50}=-\frac{750}{-50}$$

-50 ga bo'lish -50 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{300}{-50}x=-\frac{750}{-50}$$

300 ni -50 ga bo'lish.

$$x^{2}-6x=-\frac{750}{-50}$$

-750 ni -50 ga bo'lish.

$$x^{2}-6x=15$$

-6 ni bo‘lish, x shartining koeffitsienti, 2 ga -3 olish uchun. Keyin, -3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-6x+\left(-3\right)^{2}=15+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=15+9$$

15 ni 9 ga qo'shish.

$$x^{2}-6x+9=24$$

x^{2}-6x+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-3\right)^{2}=24$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{24}$$

Qisqartirish.

$$x-3=2\sqrt{6} x-3=-2\sqrt{6}$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{6}+3 x=3-2\sqrt{6}$$