450=x(100-2x) eching

$$450=x(100-2x)$$

x uchun yechish

$$x=5$$

$$x=45$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga 100-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$450=100x-2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$100x-2x^{2}=450$$

Ikkala tarafdan 450 ni ayirish.

$$100x-2x^{2}-450=0$$

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.

$$-2x^{2}+100x-450=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 100 ni b va -450 ni c bilan almashtiring.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

100 kvadratini chiqarish.

$$x=\frac{-100±\sqrt{10000-4\left(-2\right)\left(-450\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000+8\left(-450\right)}}{2\left(-2\right)}$$

8 ni -450 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000-3600}}{2\left(-2\right)}$$

10000 ni -3600 ga qo'shish.

$$x=\frac{-100±\sqrt{6400}}{2\left(-2\right)}$$

6400 ning kvadrat ildizini chiqarish.

$$x=\frac{-100±80}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-100±80}{-4}$$

x=\frac{-100±80}{-4} tenglamasini yeching, bunda ± musbat. -100 ni 80 ga qo'shish.

$$x=-\frac{20}{-4}$$

-20 ni -4 ga bo'lish.

$$x=5$$

x=\frac{-100±80}{-4} tenglamasini yeching, bunda ± manfiy. -100 dan 80 ni ayirish.

$$x=-\frac{180}{-4}$$

-180 ni -4 ga bo'lish.

$$x=45$$

Tenglama yechildi.

$$x=5 x=45$$

Kvadratni yakunlash bo'yicha qadamlar

x ga 100-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$450=100x-2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$100x-2x^{2}=450$$

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

$$-2x^{2}+100x=450$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+100x}{-2}=\frac{450}{-2}$$

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{100}{-2}x=\frac{450}{-2}$$

100 ni -2 ga bo'lish.

$$x^{2}-50x=\frac{450}{-2}$$

450 ni -2 ga bo'lish.

$$x^{2}-50x=-225$$

-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.

$$x^{2}-50x+\left(-25\right)^{2}=-225+\left(-25\right)^{2}$$

-25 kvadratini chiqarish.

$$x^{2}-50x+625=-225+625$$

-225 ni 625 ga qo'shish.

$$x^{2}-50x+625=400$$

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.

$$\left(x-25\right)^{2}=400$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{400}$$

Qisqartirish.

$$x-25=20 x-25=-20$$

25 ni tenglamaning ikkala tarafiga qo'shish.

$$x=45 x=5$$