x^2-37x+259=0 eching

$${ x }^{ 2 } -37x+259=0$$

x uchun yechish

$$x = \frac{3 \sqrt{37} + 37}{2} \approx 27,624143795$$

$$x = \frac{37 - 3 \sqrt{37}}{2} \approx 9,375856205$$

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.

$$x^{2}-37x+259=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -37 ni b va 259 ni c bilan almashtiring.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 259}}{2}$$

-37 kvadratini chiqarish.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\times 259}}{2}$$

-4 ni 259 marotabaga ko'paytirish.

$$x=\frac{-\left(-37\right)±\sqrt{1369-1036}}{2}$$

1369 ni -1036 ga qo'shish.

$$x=\frac{-\left(-37\right)±\sqrt{333}}{2}$$

333 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-37\right)±3\sqrt{37}}{2}$$

-37 ning teskarisi 37 ga teng.

$$x=\frac{37±3\sqrt{37}}{2}$$

x=\frac{37±3\sqrt{37}}{2} tenglamasini yeching, bunda ± musbat. 37 ni 3\sqrt{37} ga qo'shish.

$$x=\frac{3\sqrt{37}+37}{2}$$

x=\frac{37±3\sqrt{37}}{2} tenglamasini yeching, bunda ± manfiy. 37 dan 3\sqrt{37} ni ayirish.

$$x=\frac{37-3\sqrt{37}}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{37}+37}{2} x=\frac{37-3\sqrt{37}}{2}$$

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.

$$x^{2}-37x+259=0$$

Tenglamaning ikkala tarafidan 259 ni ayirish.

$$x^{2}-37x+259-259=-259$$

O‘zidan 259 ayirilsa 0 qoladi.

$$x^{2}-37x=-259$$

-37 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{37}{2} olish uchun. Keyin, -\frac{37}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=-259+\left(-\frac{37}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{37}{2} kvadratini chiqarish.

$$x^{2}-37x+\frac{1369}{4}=-259+\frac{1369}{4}$$

-259 ni \frac{1369}{4} ga qo'shish.

$$x^{2}-37x+\frac{1369}{4}=\frac{333}{4}$$

x^{2}-37x+\frac{1369}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{37}{2}\right)^{2}=\frac{333}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{37}{2}\right)^{2}}=\sqrt{\frac{333}{4}}$$

Qisqartirish.

$$x-\frac{37}{2}=\frac{3\sqrt{37}}{2} x-\frac{37}{2}=-\frac{3\sqrt{37}}{2}$$

\frac{37}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{37}+37}{2} x=\frac{37-3\sqrt{37}}{2}$$