100=x(30-2x) eching

$$100=x(30-2x)$$

x uchun yechish

$$x=5$$

$$x=10$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$100=30x-2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$30x-2x^{2}=100$$

Ikkala tarafdan 100 ni ayirish.

$$30x-2x^{2}-100=0$$

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.

$$-2x^{2}+30x-100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 30 ni b va -100 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\left(-2\right)\left(-100\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+8\left(-100\right)}}{2\left(-2\right)}$$

8 ni -100 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-800}}{2\left(-2\right)}$$

900 ni -800 ga qo'shish.

$$x=\frac{-30±\sqrt{100}}{2\left(-2\right)}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±10}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-30±10}{-4}$$

x=\frac{-30±10}{-4} tenglamasini yeching, bunda ± musbat. -30 ni 10 ga qo'shish.

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

-20 ni -4 ga bo'lish.

$$x=5$$

x=\frac{-30±10}{-4} tenglamasini yeching, bunda ± manfiy. -30 dan 10 ni ayirish.

$$x=-\frac{40}{-4}$$

-40 ni -4 ga bo'lish.

$$x=10$$

Tenglama yechildi.

$$x=5 x=10$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$100=30x-2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$30x-2x^{2}=100$$

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

$$-2x^{2}+30x=100$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+30x}{-2}=\frac{100}{-2}$$

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

$$x^{2}+\frac{30}{-2}x=\frac{100}{-2}$$

30 ni -2 ga bo'lish.

$$x^{2}-15x=\frac{100}{-2}$$

100 ni -2 ga bo'lish.

$$x^{2}-15x=-50$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

-50 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

x^{2}-15x+\frac{225}{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.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=5$$