{left(05sqrt{3}xright)}^2+left(5-15xright)^2=left(1+xright)^2 eching

$${ \left(05 \sqrt{ 3 } x \right) }^{ 2 } + { \left(5-15x \right) }^{ 2 } = { \left(1+x \right) }^{ 2 }$$

x uchun yechish

$$x=\frac{1}{4}=0,25$$

$$x=\frac{3}{7}\approx 0,428571429$$

Kvadrat tenglamalaridan foydalanish qoidalari

0 hosil qilish uchun 0 va 5 ni ko'paytirish.

$$\left(0\sqrt{3}x\right)^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

$$0^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

2 daraja ko‘rsatkichini 0 ga hisoblang va 0 ni qiymatni oling.

$$0+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5-15x\right)^{2} kengaytirilishi uchun ishlating.

$$0+25-150x+225x^{2}=\left(1+x\right)^{2}$$

25 olish uchun 0 va 25'ni qo'shing.

$$25-150x+225x^{2}=\left(1+x\right)^{2}$$

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

$$25-150x+225x^{2}=1+2x+x^{2}$$

Ikkala tarafdan 1 ni ayirish.

$$25-150x+225x^{2}-1=2x+x^{2}$$

24 olish uchun 25 dan 1 ni ayirish.

$$24-150x+225x^{2}=2x+x^{2}$$

Ikkala tarafdan 2x ni ayirish.

$$24-150x+225x^{2}-2x=x^{2}$$

-152x ni olish uchun -150x va -2x ni birlashtirish.

$$24-152x+225x^{2}=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$24-152x+225x^{2}-x^{2}=0$$

224x^{2} ni olish uchun 225x^{2} va -x^{2} ni birlashtirish.

$$24-152x+224x^{2}=0$$

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.

$$224x^{2}-152x+24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 224 ni a, -152 ni b va 24 ni c bilan almashtiring.

$$x=\frac{-\left(-152\right)±\sqrt{\left(-152\right)^{2}-4\times 224\times 24}}{2\times 224}$$

-152 kvadratini chiqarish.

$$x=\frac{-\left(-152\right)±\sqrt{23104-4\times 224\times 24}}{2\times 224}$$

-4 ni 224 marotabaga ko'paytirish.

$$x=\frac{-\left(-152\right)±\sqrt{23104-896\times 24}}{2\times 224}$$

-896 ni 24 marotabaga ko'paytirish.

$$x=\frac{-\left(-152\right)±\sqrt{23104-21504}}{2\times 224}$$

23104 ni -21504 ga qo'shish.

$$x=\frac{-\left(-152\right)±\sqrt{1600}}{2\times 224}$$

1600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-152\right)±40}{2\times 224}$$

-152 ning teskarisi 152 ga teng.

$$x=\frac{152±40}{2\times 224}$$

2 ni 224 marotabaga ko'paytirish.

$$x=\frac{152±40}{448}$$

x=\frac{152±40}{448} tenglamasini yeching, bunda ± musbat. 152 ni 40 ga qo'shish.

$$x=\frac{192}{448}$$

\frac{192}{448} ulushini 64 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{3}{7}$$

x=\frac{152±40}{448} tenglamasini yeching, bunda ± manfiy. 152 dan 40 ni ayirish.

$$x=\frac{112}{448}$$

\frac{112}{448} ulushini 112 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{4}$$

Tenglama yechildi.

$$x=\frac{3}{7} x=\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

0 hosil qilish uchun 0 va 5 ni ko'paytirish.

$$\left(0\sqrt{3}x\right)^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

Har qanday sonni nolga ko‘paytirsangiz, nol chiqadi.

$$0^{2}+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

2 daraja ko‘rsatkichini 0 ga hisoblang va 0 ni qiymatni oling.

$$0+\left(5-15x\right)^{2}=\left(1+x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5-15x\right)^{2} kengaytirilishi uchun ishlating.

$$0+25-150x+225x^{2}=\left(1+x\right)^{2}$$

25 olish uchun 0 va 25'ni qo'shing.

$$25-150x+225x^{2}=\left(1+x\right)^{2}$$

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

$$25-150x+225x^{2}=1+2x+x^{2}$$

Ikkala tarafdan 2x ni ayirish.

$$25-150x+225x^{2}-2x=1+x^{2}$$

-152x ni olish uchun -150x va -2x ni birlashtirish.

$$25-152x+225x^{2}=1+x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$25-152x+225x^{2}-x^{2}=1$$

224x^{2} ni olish uchun 225x^{2} va -x^{2} ni birlashtirish.

$$25-152x+224x^{2}=1$$

Ikkala tarafdan 25 ni ayirish.

$$-152x+224x^{2}=1-25$$

-24 olish uchun 1 dan 25 ni ayirish.

$$-152x+224x^{2}=-24$$

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

$$224x^{2}-152x=-24$$

Ikki tarafini 224 ga bo‘ling.

$$\frac{224x^{2}-152x}{224}=-\frac{24}{224}$$

224 ga bo'lish 224 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{152}{224}\right)x=-\frac{24}{224}$$

\frac{-152}{224} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{19}{28}x=-\frac{24}{224}$$

\frac{-24}{224} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{19}{28}x=-\frac{3}{28}$$

-\frac{19}{28} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{56} olish uchun. Keyin, -\frac{19}{56} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{28}x+\left(-\frac{19}{56}\right)^{2}=-\frac{3}{28}+\left(-\frac{19}{56}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{56} kvadratini chiqarish.

$$x^{2}-\frac{19}{28}x+\frac{361}{3136}=-\frac{3}{28}+\frac{361}{3136}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{28} ni \frac{361}{3136} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{19}{28}x+\frac{361}{3136}=\frac{25}{3136}$$

x^{2}-\frac{19}{28}x+\frac{361}{3136} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{19}{56}\right)^{2}=\frac{25}{3136}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{56}\right)^{2}}=\sqrt{\frac{25}{3136}}$$

Qisqartirish.

$$x-\frac{19}{56}=\frac{5}{56} x-\frac{19}{56}=-\frac{5}{56}$$

\frac{19}{56} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3}{7} x=\frac{1}{4}$$