{left(1/4xright)}^2+{left(80/4-1/4xright)}^2=200 eching

$${ \left( \frac{ 1 }{ 4 } x \right) }^{ 2 } + { \left( \frac{ 80 }{ 4 } - \frac{ 1 }{ 4 } x \right) }^{ 2 } =200$$

x uchun yechish

$$x=40$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(\frac{1}{4}x\right)^{2} ni kengaytirish.

$$\left(\frac{1}{4}\right)^{2}x^{2}+\left(\frac{80}{4}-\frac{1}{4}x\right)^{2}=200$$

2 daraja ko‘rsatkichini \frac{1}{4} ga hisoblang va \frac{1}{16} ni qiymatni oling.

$$\frac{1}{16}x^{2}+\left(\frac{80}{4}-\frac{1}{4}x\right)^{2}=200$$

20 ni olish uchun 80 ni 4 ga bo‘ling.

$$\frac{1}{16}x^{2}+\left(20-\frac{1}{4}x\right)^{2}=200$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(20-\frac{1}{4}x\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{1}{16}x^{2}+400-10x+\frac{1}{16}x^{2}=200$$

\frac{1}{8}x^{2} ni olish uchun \frac{1}{16}x^{2} va \frac{1}{16}x^{2} ni birlashtirish.

$$\frac{1}{8}x^{2}+400-10x=200$$

Ikkala tarafdan 200 ni ayirish.

$$\frac{1}{8}x^{2}+400-10x-200=0$$

200 olish uchun 400 dan 200 ni ayirish.

$$\frac{1}{8}x^{2}+200-10x=0$$

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.

$$\frac{1}{8}x^{2}-10x+200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{8} ni a, -10 ni b va 200 ni c bilan almashtiring.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times \frac{1}{8}\times 200}}{2\times \frac{1}{8}}$$

-10 kvadratini chiqarish.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times \frac{1}{8}\times 200}}{2\times \frac{1}{8}}$$

-4 ni \frac{1}{8} marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-\frac{1}{2}\times 200}}{2\times \frac{1}{8}}$$

-\frac{1}{2} ni 200 marotabaga ko'paytirish.

$$x=\frac{-\left(-10\right)±\sqrt{100-100}}{2\times \frac{1}{8}}$$

100 ni -100 ga qo'shish.

$$x=\frac{-\left(-10\right)±\sqrt{0}}{2\times \frac{1}{8}}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-10}{2\times \frac{1}{8}}$$

-10 ning teskarisi 10 ga teng.

$$x=\frac{10}{2\times \frac{1}{8}}$$

2 ni \frac{1}{8} marotabaga ko'paytirish.

$$x=\frac{10}{\frac{1}{4}}$$

10 ni \frac{1}{4} ga bo'lish 10 ga k'paytirish \frac{1}{4} ga qaytarish.

$$x=40$$

Kvadratni yakunlash bo'yicha qadamlar

\left(\frac{1}{4}x\right)^{2} ni kengaytirish.

$$\left(\frac{1}{4}\right)^{2}x^{2}+\left(\frac{80}{4}-\frac{1}{4}x\right)^{2}=200$$

2 daraja ko‘rsatkichini \frac{1}{4} ga hisoblang va \frac{1}{16} ni qiymatni oling.

$$\frac{1}{16}x^{2}+\left(\frac{80}{4}-\frac{1}{4}x\right)^{2}=200$$

20 ni olish uchun 80 ni 4 ga bo‘ling.

$$\frac{1}{16}x^{2}+\left(20-\frac{1}{4}x\right)^{2}=200$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(20-\frac{1}{4}x\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{1}{16}x^{2}+400-10x+\frac{1}{16}x^{2}=200$$

\frac{1}{8}x^{2} ni olish uchun \frac{1}{16}x^{2} va \frac{1}{16}x^{2} ni birlashtirish.

$$\frac{1}{8}x^{2}+400-10x=200$$

Ikkala tarafdan 400 ni ayirish.

$$\frac{1}{8}x^{2}-10x=200-400$$

-200 olish uchun 200 dan 400 ni ayirish.

$$\frac{1}{8}x^{2}-10x=-200$$

Ikkala tarafini 8 ga ko‘paytiring.

$$\frac{\frac{1}{8}x^{2}-10x}{\frac{1}{8}}=-\frac{200}{\frac{1}{8}}$$

\frac{1}{8} ga bo'lish \frac{1}{8} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{10}{\frac{1}{8}}\right)x=-\frac{200}{\frac{1}{8}}$$

-10 ni \frac{1}{8} ga bo'lish -10 ga k'paytirish \frac{1}{8} ga qaytarish.

$$x^{2}-80x=-\frac{200}{\frac{1}{8}}$$

-200 ni \frac{1}{8} ga bo'lish -200 ga k'paytirish \frac{1}{8} ga qaytarish.

$$x^{2}-80x=-1600$$

-80 ni bo‘lish, x shartining koeffitsienti, 2 ga -40 olish uchun. Keyin, -40 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-80x+\left(-40\right)^{2}=-1600+\left(-40\right)^{2}$$

-40 kvadratini chiqarish.

$$x^{2}-80x+1600=-1600+1600$$

-1600 ni 1600 ga qo'shish.

$$x^{2}-80x+1600=0$$

x^{2}-80x+1600 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-40\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-40=0 x-40=0$$

40 ni tenglamaning ikkala tarafiga qo'shish.

$$x=40 x=40$$

Tenglama yechildi. Yechimlar bir xil.

$$x=40$$