27x^2+59x-21=0 eching

$$27 { x }^{ 2 } +59x-21=0$$

x uchun yechish

$$x=\frac{\sqrt{5749}-59}{54}\approx 0,311521488$$

$$x=\frac{-\sqrt{5749}-59}{54}\approx -2,496706673$$

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.

$$27x^{2}+59x-21=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 27 ni a, 59 ni b va -21 ni c bilan almashtiring.

$$x=\frac{-59±\sqrt{59^{2}-4\times 27\left(-21\right)}}{2\times 27}$$

59 kvadratini chiqarish.

$$x=\frac{-59±\sqrt{3481-4\times 27\left(-21\right)}}{2\times 27}$$

-4 ni 27 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{3481-108\left(-21\right)}}{2\times 27}$$

-108 ni -21 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{3481+2268}}{2\times 27}$$

3481 ni 2268 ga qo'shish.

$$x=\frac{-59±\sqrt{5749}}{2\times 27}$$

2 ni 27 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{5749}}{54}$$

x=\frac{-59±\sqrt{5749}}{54} tenglamasini yeching, bunda ± musbat. -59 ni \sqrt{5749} ga qo'shish.

$$x=\frac{\sqrt{5749}-59}{54}$$

x=\frac{-59±\sqrt{5749}}{54} tenglamasini yeching, bunda ± manfiy. -59 dan \sqrt{5749} ni ayirish.

$$x=\frac{-\sqrt{5749}-59}{54}$$

Tenglama yechildi.

$$x=\frac{\sqrt{5749}-59}{54} x=\frac{-\sqrt{5749}-59}{54}$$

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.

$$27x^{2}+59x-21=0$$

21 ni tenglamaning ikkala tarafiga qo'shish.

$$27x^{2}+59x-21-\left(-21\right)=-\left(-21\right)$$

O‘zidan -21 ayirilsa 0 qoladi.

$$27x^{2}+59x=-\left(-21\right)$$

0 dan -21 ni ayirish.

$$27x^{2}+59x=21$$

Ikki tarafini 27 ga bo‘ling.

$$\frac{27x^{2}+59x}{27}=\frac{21}{27}$$

27 ga bo'lish 27 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{59}{27}x=\frac{21}{27}$$

\frac{21}{27} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{59}{27}x=\frac{7}{9}$$

\frac{59}{27} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{59}{54} olish uchun. Keyin, \frac{59}{54} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{59}{27}x+\left(\frac{59}{54}\right)^{2}=\frac{7}{9}+\left(\frac{59}{54}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{59}{54} kvadratini chiqarish.

$$x^{2}+\frac{59}{27}x+\frac{3481}{2916}=\frac{7}{9}+\frac{3481}{2916}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{9} ni \frac{3481}{2916} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{59}{27}x+\frac{3481}{2916}=\frac{5749}{2916}$$

x^{2}+\frac{59}{27}x+\frac{3481}{2916} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{59}{54}\right)^{2}=\frac{5749}{2916}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{59}{54}\right)^{2}}=\sqrt{\frac{5749}{2916}}$$

Qisqartirish.

$$x+\frac{59}{54}=\frac{\sqrt{5749}}{54} x+\frac{59}{54}=-\frac{\sqrt{5749}}{54}$$

Tenglamaning ikkala tarafidan \frac{59}{54} ni ayirish.

$$x=\frac{\sqrt{5749}-59}{54} x=\frac{-\sqrt{5749}-59}{54}$$