(x-159)(359-x)=20(x+87) eching

$$(x-159)(359-x)=20(x+87)$$

x uchun yechish

$$x=2\sqrt{795}+249\approx 305,391488719$$

$$x=249-2\sqrt{795}\approx 192,608511281$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-159 ga 359-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$518x-x^{2}-57081=20\left(x+87\right)$$

20 ga x+87 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$518x-x^{2}-57081=20x+1740$$

Ikkala tarafdan 20x ni ayirish.

$$518x-x^{2}-57081-20x=1740$$

498x ni olish uchun 518x va -20x ni birlashtirish.

$$498x-x^{2}-57081=1740$$

Ikkala tarafdan 1740 ni ayirish.

$$498x-x^{2}-57081-1740=0$$

-58821 olish uchun -57081 dan 1740 ni ayirish.

$$498x-x^{2}-58821=0$$

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.

$$-x^{2}+498x-58821=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 498 ni b va -58821 ni c bilan almashtiring.

$$x=\frac{-498±\sqrt{498^{2}-4\left(-1\right)\left(-58821\right)}}{2\left(-1\right)}$$

498 kvadratini chiqarish.

$$x=\frac{-498±\sqrt{248004-4\left(-1\right)\left(-58821\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-498±\sqrt{248004+4\left(-58821\right)}}{2\left(-1\right)}$$

4 ni -58821 marotabaga ko'paytirish.

$$x=\frac{-498±\sqrt{248004-235284}}{2\left(-1\right)}$$

248004 ni -235284 ga qo'shish.

$$x=\frac{-498±\sqrt{12720}}{2\left(-1\right)}$$

12720 ning kvadrat ildizini chiqarish.

$$x=\frac{-498±4\sqrt{795}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-498±4\sqrt{795}}{-2}$$

x=\frac{-498±4\sqrt{795}}{-2} tenglamasini yeching, bunda ± musbat. -498 ni 4\sqrt{795} ga qo'shish.

$$x=\frac{4\sqrt{795}-498}{-2}$$

-498+4\sqrt{795} ni -2 ga bo'lish.

$$x=249-2\sqrt{795}$$

x=\frac{-498±4\sqrt{795}}{-2} tenglamasini yeching, bunda ± manfiy. -498 dan 4\sqrt{795} ni ayirish.

$$x=\frac{-4\sqrt{795}-498}{-2}$$

-498-4\sqrt{795} ni -2 ga bo'lish.

$$x=2\sqrt{795}+249$$

Tenglama yechildi.

$$x=249-2\sqrt{795} x=2\sqrt{795}+249$$

Kvadratni yakunlash bo'yicha qadamlar

x-159 ga 359-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$518x-x^{2}-57081=20\left(x+87\right)$$

20 ga x+87 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$518x-x^{2}-57081=20x+1740$$

Ikkala tarafdan 20x ni ayirish.

$$518x-x^{2}-57081-20x=1740$$

498x ni olish uchun 518x va -20x ni birlashtirish.

$$498x-x^{2}-57081=1740$$

57081 ni ikki tarafga qo’shing.

$$498x-x^{2}=1740+57081$$

58821 olish uchun 1740 va 57081'ni qo'shing.

$$498x-x^{2}=58821$$

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

$$-x^{2}+498x=58821$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+498x}{-1}=\frac{58821}{-1}$$

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

$$x^{2}+\frac{498}{-1}x=\frac{58821}{-1}$$

498 ni -1 ga bo'lish.

$$x^{2}-498x=\frac{58821}{-1}$$

58821 ni -1 ga bo'lish.

$$x^{2}-498x=-58821$$

-498 ni bo‘lish, x shartining koeffitsienti, 2 ga -249 olish uchun. Keyin, -249 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-498x+\left(-249\right)^{2}=-58821+\left(-249\right)^{2}$$

-249 kvadratini chiqarish.

$$x^{2}-498x+62001=-58821+62001$$

-58821 ni 62001 ga qo'shish.

$$x^{2}-498x+62001=3180$$

x^{2}-498x+62001 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-249\right)^{2}=3180$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-249\right)^{2}}=\sqrt{3180}$$

Qisqartirish.

$$x-249=2\sqrt{795} x-249=-2\sqrt{795}$$

249 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{795}+249 x=249-2\sqrt{795}$$