1/4x^2-25x+25=0 eching

$$\frac{ 1 }{ 4 } { x }^{ 2 } -25x+25=0$$

x uchun yechish

$$x=20\sqrt{6}+50\approx 98,989794856$$

$$x=50-20\sqrt{6}\approx 1,010205144$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$\frac{1}{4}x^{2}-25x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{4} ni a, -25 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times \frac{1}{4}\times 25}}{2\times \frac{1}{4}}$$

-25 kvadratini chiqarish.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times \frac{1}{4}\times 25}}{2\times \frac{1}{4}}$$

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

$$x=\frac{-\left(-25\right)±\sqrt{625-25}}{2\times \frac{1}{4}}$$

625 ni -25 ga qo'shish.

$$x=\frac{-\left(-25\right)±\sqrt{600}}{2\times \frac{1}{4}}$$

600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-25\right)±10\sqrt{6}}{2\times \frac{1}{4}}$$

-25 ning teskarisi 25 ga teng.

$$x=\frac{25±10\sqrt{6}}{2\times \frac{1}{4}}$$

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

$$x=\frac{25±10\sqrt{6}}{\frac{1}{2}}$$

x=\frac{25±10\sqrt{6}}{\frac{1}{2}} tenglamasini yeching, bunda ± musbat. 25 ni 10\sqrt{6} ga qo'shish.

$$x=\frac{10\sqrt{6}+25}{\frac{1}{2}}$$

25+10\sqrt{6} ni \frac{1}{2} ga bo'lish 25+10\sqrt{6} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x=20\sqrt{6}+50$$

x=\frac{25±10\sqrt{6}}{\frac{1}{2}} tenglamasini yeching, bunda ± manfiy. 25 dan 10\sqrt{6} ni ayirish.

$$x=\frac{25-10\sqrt{6}}{\frac{1}{2}}$$

25-10\sqrt{6} ni \frac{1}{2} ga bo'lish 25-10\sqrt{6} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x=50-20\sqrt{6}$$

Tenglama yechildi.

$$x=20\sqrt{6}+50 x=50-20\sqrt{6}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\frac{1}{4}x^{2}-25x+25=0$$

Tenglamaning ikkala tarafidan 25 ni ayirish.

$$\frac{1}{4}x^{2}-25x+25-25=-25$$

O‘zidan 25 ayirilsa 0 qoladi.

$$\frac{1}{4}x^{2}-25x=-25$$

Ikkala tarafini 4 ga ko‘paytiring.

$$\frac{\frac{1}{4}x^{2}-25x}{\frac{1}{4}}=-\frac{25}{\frac{1}{4}}$$

\frac{1}{4} ga bo'lish \frac{1}{4} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{25}{\frac{1}{4}}\right)x=-\frac{25}{\frac{1}{4}}$$

-25 ni \frac{1}{4} ga bo'lish -25 ga k'paytirish \frac{1}{4} ga qaytarish.

$$x^{2}-100x=-\frac{25}{\frac{1}{4}}$$

-25 ni \frac{1}{4} ga bo'lish -25 ga k'paytirish \frac{1}{4} ga qaytarish.

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

-100 ni bo‘lish, x shartining koeffitsienti, 2 ga -50 olish uchun. Keyin, -50 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-100x+\left(-50\right)^{2}=-100+\left(-50\right)^{2}$$

-50 kvadratini chiqarish.

$$x^{2}-100x+2500=-100+2500$$

-100 ni 2500 ga qo'shish.

$$x^{2}-100x+2500=2400$$

x^{2}-100x+2500 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-50\right)^{2}=2400$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2400}$$

Qisqartirish.

$$x-50=20\sqrt{6} x-50=-20\sqrt{6}$$

50 ni tenglamaning ikkala tarafiga qo'shish.

$$x=20\sqrt{6}+50 x=50-20\sqrt{6}$$