2x^2-34x+20=0 eching

$$2 { x }^{ 2 } -34x+20=0$$

x uchun yechish

$$x = \frac{\sqrt{249} + 17}{2} \approx 16,389866919$$

$$x=\frac{17-\sqrt{249}}{2}\approx 0,610133081$$

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.

$$2x^{2}-34x+20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -34 ni b va 20 ni c bilan almashtiring.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 20}}{2\times 2}$$

-34 kvadratini chiqarish.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 20}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 20}}{2\times 2}$$

-8 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-34\right)±\sqrt{1156-160}}{2\times 2}$$

1156 ni -160 ga qo'shish.

$$x=\frac{-\left(-34\right)±\sqrt{996}}{2\times 2}$$

996 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-34\right)±2\sqrt{249}}{2\times 2}$$

-34 ning teskarisi 34 ga teng.

$$x=\frac{34±2\sqrt{249}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{34±2\sqrt{249}}{4}$$

x=\frac{34±2\sqrt{249}}{4} tenglamasini yeching, bunda ± musbat. 34 ni 2\sqrt{249} ga qo'shish.

$$x=\frac{2\sqrt{249}+34}{4}$$

34+2\sqrt{249} ni 4 ga bo'lish.

$$x=\frac{\sqrt{249}+17}{2}$$

x=\frac{34±2\sqrt{249}}{4} tenglamasini yeching, bunda ± manfiy. 34 dan 2\sqrt{249} ni ayirish.

$$x=\frac{34-2\sqrt{249}}{4}$$

34-2\sqrt{249} ni 4 ga bo'lish.

$$x=\frac{17-\sqrt{249}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{249}+17}{2} x=\frac{17-\sqrt{249}}{2}$$

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.

$$2x^{2}-34x+20=0$$

Tenglamaning ikkala tarafidan 20 ni ayirish.

$$2x^{2}-34x+20-20=-20$$

O‘zidan 20 ayirilsa 0 qoladi.

$$2x^{2}-34x=-20$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-34x}{2}=-\frac{20}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{20}{2}$$

-34 ni 2 ga bo'lish.

$$x^{2}-17x=-\frac{20}{2}$$

-20 ni 2 ga bo'lish.

$$x^{2}-17x=-10$$

-17 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{17}{2} olish uchun. Keyin, -\frac{17}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-10+\left(-\frac{17}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{17}{2} kvadratini chiqarish.

$$x^{2}-17x+\frac{289}{4}=-10+\frac{289}{4}$$

-10 ni \frac{289}{4} ga qo'shish.

$$x^{2}-17x+\frac{289}{4}=\frac{249}{4}$$

x^{2}-17x+\frac{289}{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.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{249}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{249}{4}}$$

Qisqartirish.

$$x-\frac{17}{2}=\frac{\sqrt{249}}{2} x-\frac{17}{2}=-\frac{\sqrt{249}}{2}$$

\frac{17}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{249}+17}{2} x=\frac{17-\sqrt{249}}{2}$$