50(1-10%)(1+x)^2=148 eching

$$50 ( 1 - 10 \% ) ( 1 + x ) ^ { 2 } = 148$$

x uchun yechish

$$x=\frac{2\sqrt{185}}{15}-1\approx 0,813529401$$

$$x=-\frac{2\sqrt{185}}{15}-1\approx -2,813529401$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{10}{100} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$50\left(1-\frac{1}{10}\right)\left(1+x\right)^{2}=148$$

\frac{9}{10} olish uchun 1 dan \frac{1}{10} ni ayirish.

$$50\times \frac{9}{10}\left(1+x\right)^{2}=148$$

45 hosil qilish uchun 50 va \frac{9}{10} ni ko'paytirish.

$$45\left(1+x\right)^{2}=148$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(1+x\right)^{2} kengaytirilishi uchun ishlating.

$$45\left(1+2x+x^{2}\right)=148$$

45 ga 1+2x+x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$45+90x+45x^{2}=148$$

Ikkala tarafdan 148 ni ayirish.

$$45+90x+45x^{2}-148=0$$

-103 olish uchun 45 dan 148 ni ayirish.

$$-103+90x+45x^{2}=0$$

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.

$$45x^{2}+90x-103=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 45 ni a, 90 ni b va -103 ni c bilan almashtiring.

$$x=\frac{-90±\sqrt{90^{2}-4\times 45\left(-103\right)}}{2\times 45}$$

90 kvadratini chiqarish.

$$x=\frac{-90±\sqrt{8100-4\times 45\left(-103\right)}}{2\times 45}$$

-4 ni 45 marotabaga ko'paytirish.

$$x=\frac{-90±\sqrt{8100-180\left(-103\right)}}{2\times 45}$$

-180 ni -103 marotabaga ko'paytirish.

$$x=\frac{-90±\sqrt{8100+18540}}{2\times 45}$$

8100 ni 18540 ga qo'shish.

$$x=\frac{-90±\sqrt{26640}}{2\times 45}$$

26640 ning kvadrat ildizini chiqarish.

$$x=\frac{-90±12\sqrt{185}}{2\times 45}$$

2 ni 45 marotabaga ko'paytirish.

$$x=\frac{-90±12\sqrt{185}}{90}$$

x=\frac{-90±12\sqrt{185}}{90} tenglamasini yeching, bunda ± musbat. -90 ni 12\sqrt{185} ga qo'shish.

$$x=\frac{12\sqrt{185}-90}{90}$$

-90+12\sqrt{185} ni 90 ga bo'lish.

$$x=\frac{2\sqrt{185}}{15}-1$$

x=\frac{-90±12\sqrt{185}}{90} tenglamasini yeching, bunda ± manfiy. -90 dan 12\sqrt{185} ni ayirish.

$$x=\frac{-12\sqrt{185}-90}{90}$$

-90-12\sqrt{185} ni 90 ga bo'lish.

$$x=-\frac{2\sqrt{185}}{15}-1$$

Tenglama yechildi.

$$x=\frac{2\sqrt{185}}{15}-1 x=-\frac{2\sqrt{185}}{15}-1$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{10}{100} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$50\left(1-\frac{1}{10}\right)\left(1+x\right)^{2}=148$$

\frac{9}{10} olish uchun 1 dan \frac{1}{10} ni ayirish.

$$50\times \frac{9}{10}\left(1+x\right)^{2}=148$$

45 hosil qilish uchun 50 va \frac{9}{10} ni ko'paytirish.

$$45\left(1+x\right)^{2}=148$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(1+x\right)^{2} kengaytirilishi uchun ishlating.

$$45\left(1+2x+x^{2}\right)=148$$

45 ga 1+2x+x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$45+90x+45x^{2}=148$$

Ikkala tarafdan 45 ni ayirish.

$$90x+45x^{2}=148-45$$

103 olish uchun 148 dan 45 ni ayirish.

$$90x+45x^{2}=103$$

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

$$45x^{2}+90x=103$$

Ikki tarafini 45 ga bo‘ling.

$$\frac{45x^{2}+90x}{45}=\frac{103}{45}$$

45 ga bo'lish 45 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{90}{45}x=\frac{103}{45}$$

90 ni 45 ga bo'lish.

$$x^{2}+2x=\frac{103}{45}$$

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+2x+1^{2}=\frac{103}{45}+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=\frac{103}{45}+1$$

\frac{103}{45} ni 1 ga qo'shish.

$$x^{2}+2x+1=\frac{148}{45}$$

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+1\right)^{2}=\frac{148}{45}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{148}{45}}$$

Qisqartirish.

$$x+1=\frac{2\sqrt{185}}{15} x+1=-\frac{2\sqrt{185}}{15}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\frac{2\sqrt{185}}{15}-1 x=-\frac{2\sqrt{185}}{15}-1$$