40x+2x(30-2x)=200 eching

$$40x+2x(30-2x)=200$$

x uchun yechish

$$x = \frac{5 \sqrt{17} + 25}{2} \approx 22,807764064$$

$$x = \frac{25 - 5 \sqrt{17}}{2} \approx 2,192235936$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x ga 30-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$40x+60x-4x^{2}=200$$

100x ni olish uchun 40x va 60x ni birlashtirish.

$$100x-4x^{2}=200$$

Ikkala tarafdan 200 ni ayirish.

$$100x-4x^{2}-200=0$$

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.

$$-4x^{2}+100x-200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 100 ni b va -200 ni c bilan almashtiring.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-200\right)}}{2\left(-4\right)}$$

100 kvadratini chiqarish.

$$x=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-200\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000+16\left(-200\right)}}{2\left(-4\right)}$$

16 ni -200 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000-3200}}{2\left(-4\right)}$$

10000 ni -3200 ga qo'shish.

$$x=\frac{-100±\sqrt{6800}}{2\left(-4\right)}$$

6800 ning kvadrat ildizini chiqarish.

$$x=\frac{-100±20\sqrt{17}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-100±20\sqrt{17}}{-8}$$

x=\frac{-100±20\sqrt{17}}{-8} tenglamasini yeching, bunda ± musbat. -100 ni 20\sqrt{17} ga qo'shish.

$$x=\frac{20\sqrt{17}-100}{-8}$$

-100+20\sqrt{17} ni -8 ga bo'lish.

$$x=\frac{25-5\sqrt{17}}{2}$$

x=\frac{-100±20\sqrt{17}}{-8} tenglamasini yeching, bunda ± manfiy. -100 dan 20\sqrt{17} ni ayirish.

$$x=\frac{-20\sqrt{17}-100}{-8}$$

-100-20\sqrt{17} ni -8 ga bo'lish.

$$x=\frac{5\sqrt{17}+25}{2}$$

Tenglama yechildi.

$$x=\frac{25-5\sqrt{17}}{2} x=\frac{5\sqrt{17}+25}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2x ga 30-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$40x+60x-4x^{2}=200$$

100x ni olish uchun 40x va 60x ni birlashtirish.

$$100x-4x^{2}=200$$

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

$$-4x^{2}+100x=200$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+100x}{-4}=\frac{200}{-4}$$

-4 ga bo'lish -4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{100}{-4}x=\frac{200}{-4}$$

100 ni -4 ga bo'lish.

$$x^{2}-25x=\frac{200}{-4}$$

200 ni -4 ga bo'lish.

$$x^{2}-25x=-50$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{25}{2} kvadratini chiqarish.

$$x^{2}-25x+\frac{625}{4}=-50+\frac{625}{4}$$

-50 ni \frac{625}{4} ga qo'shish.

$$x^{2}-25x+\frac{625}{4}=\frac{425}{4}$$

x^{2}-25x+\frac{625}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{425}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{425}{4}}$$

Qisqartirish.

$$x-\frac{25}{2}=\frac{5\sqrt{17}}{2} x-\frac{25}{2}=-\frac{5\sqrt{17}}{2}$$

\frac{25}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5\sqrt{17}+25}{2} x=\frac{25-5\sqrt{17}}{2}$$