2x(38-2x)=180 eching

$$2x(38-2x)=180$$

x uchun yechish

$$x = \frac{\sqrt{181} + 19}{2} \approx 16,226812024$$

$$x = \frac{19 - \sqrt{181}}{2} \approx 2,773187976$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x ga 38-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$76x-4x^{2}=180$$

Ikkala tarafdan 180 ni ayirish.

$$76x-4x^{2}-180=0$$

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.

$$-4x^{2}+76x-180=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 76 ni b va -180 ni c bilan almashtiring.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-4\right)\left(-180\right)}}{2\left(-4\right)}$$

76 kvadratini chiqarish.

$$x=\frac{-76±\sqrt{5776-4\left(-4\right)\left(-180\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-76±\sqrt{5776+16\left(-180\right)}}{2\left(-4\right)}$$

16 ni -180 marotabaga ko'paytirish.

$$x=\frac{-76±\sqrt{5776-2880}}{2\left(-4\right)}$$

5776 ni -2880 ga qo'shish.

$$x=\frac{-76±\sqrt{2896}}{2\left(-4\right)}$$

2896 ning kvadrat ildizini chiqarish.

$$x=\frac{-76±4\sqrt{181}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-76±4\sqrt{181}}{-8}$$

x=\frac{-76±4\sqrt{181}}{-8} tenglamasini yeching, bunda ± musbat. -76 ni 4\sqrt{181} ga qo'shish.

$$x=\frac{4\sqrt{181}-76}{-8}$$

-76+4\sqrt{181} ni -8 ga bo'lish.

$$x=\frac{19-\sqrt{181}}{2}$$

x=\frac{-76±4\sqrt{181}}{-8} tenglamasini yeching, bunda ± manfiy. -76 dan 4\sqrt{181} ni ayirish.

$$x=\frac{-4\sqrt{181}-76}{-8}$$

-76-4\sqrt{181} ni -8 ga bo'lish.

$$x=\frac{\sqrt{181}+19}{2}$$

Tenglama yechildi.

$$x=\frac{19-\sqrt{181}}{2} x=\frac{\sqrt{181}+19}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2x ga 38-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$76x-4x^{2}=180$$

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

$$-4x^{2}+76x=180$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+76x}{-4}=\frac{180}{-4}$$

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

$$x^{2}+\frac{76}{-4}x=\frac{180}{-4}$$

76 ni -4 ga bo'lish.

$$x^{2}-19x=\frac{180}{-4}$$

180 ni -4 ga bo'lish.

$$x^{2}-19x=-45$$

-19 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{2} olish uchun. Keyin, -\frac{19}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-45+\left(-\frac{19}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{2} kvadratini chiqarish.

$$x^{2}-19x+\frac{361}{4}=-45+\frac{361}{4}$$

-45 ni \frac{361}{4} ga qo'shish.

$$x^{2}-19x+\frac{361}{4}=\frac{181}{4}$$

x^{2}-19x+\frac{361}{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.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{181}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{181}{4}}$$

Qisqartirish.

$$x-\frac{19}{2}=\frac{\sqrt{181}}{2} x-\frac{19}{2}=-\frac{\sqrt{181}}{2}$$

\frac{19}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{181}+19}{2} x=\frac{19-\sqrt{181}}{2}$$