(k+1/4)^2-1/16-1/5=0 eching

$$( k + \frac { 1 } { 4 } ) ^ { 2 } - \frac { 1 } { 16 } - \frac { 1 } { 5 } = 0$$

k uchun yechish

$$k=\frac{\sqrt{105}}{20}-\frac{1}{4}\approx 0,262347538$$

$$k=-\frac{\sqrt{105}}{20}-\frac{1}{4}\approx -0,762347538$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$k^{2}+\frac{1}{2}k+\frac{1}{16}-\frac{1}{16}-\frac{1}{5}=0$$

0 olish uchun \frac{1}{16} dan \frac{1}{16} ni ayirish.

$$k^{2}+\frac{1}{2}k-\frac{1}{5}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, \frac{1}{2} ni b va -\frac{1}{5} ni c bilan almashtiring.

$$k=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\left(-\frac{1}{5}\right)}}{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$k=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\left(-\frac{1}{5}\right)}}{2}$$

-4 ni -\frac{1}{5} marotabaga ko'paytirish.

$$k=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+\frac{4}{5}}}{2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{4} ni \frac{4}{5} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k=\frac{-\frac{1}{2}±\sqrt{\frac{21}{20}}}{2}$$

\frac{21}{20} ning kvadrat ildizini chiqarish.

$$k=\frac{-\frac{1}{2}±\frac{\sqrt{105}}{10}}{2}$$

k=\frac{-\frac{1}{2}±\frac{\sqrt{105}}{10}}{2} tenglamasini yeching, bunda ± musbat. -\frac{1}{2} ni \frac{\sqrt{105}}{10} ga qo'shish.

$$k=\frac{\frac{\sqrt{105}}{10}-\frac{1}{2}}{2}$$

-\frac{1}{2}+\frac{\sqrt{105}}{10} ni 2 ga bo'lish.

$$k=\frac{\sqrt{105}}{20}-\frac{1}{4}$$

k=\frac{-\frac{1}{2}±\frac{\sqrt{105}}{10}}{2} tenglamasini yeching, bunda ± manfiy. -\frac{1}{2} dan \frac{\sqrt{105}}{10} ni ayirish.

$$k=\frac{-\frac{\sqrt{105}}{10}-\frac{1}{2}}{2}$$

-\frac{1}{2}-\frac{\sqrt{105}}{10} ni 2 ga bo'lish.

$$k=-\frac{\sqrt{105}}{20}-\frac{1}{4}$$

Tenglama yechildi.

$$k=\frac{\sqrt{105}}{20}-\frac{1}{4} k=-\frac{\sqrt{105}}{20}-\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$k^{2}+\frac{1}{2}k+\frac{1}{16}-\frac{1}{16}-\frac{1}{5}=0$$

0 olish uchun \frac{1}{16} dan \frac{1}{16} ni ayirish.

$$k^{2}+\frac{1}{2}k-\frac{1}{5}=0$$

\frac{1}{5} ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$k^{2}+\frac{1}{2}k=\frac{1}{5}$$

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}+\frac{1}{2}k+\left(\frac{1}{4}\right)^{2}=\frac{1}{5}+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$k^{2}+\frac{1}{2}k+\frac{1}{16}=\frac{1}{5}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{5} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}+\frac{1}{2}k+\frac{1}{16}=\frac{21}{80}$$

k^{2}+\frac{1}{2}k+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k+\frac{1}{4}\right)^{2}=\frac{21}{80}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k+\frac{1}{4}\right)^{2}}=\sqrt{\frac{21}{80}}$$

Qisqartirish.

$$k+\frac{1}{4}=\frac{\sqrt{105}}{20} k+\frac{1}{4}=-\frac{\sqrt{105}}{20}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$k=\frac{\sqrt{105}}{20}-\frac{1}{4} k=-\frac{\sqrt{105}}{20}-\frac{1}{4}$$