2x^2-3x+8=50 eching

$$2 x ^ { 2 } - 3 x + 8 = 50$$

x uchun yechish

$$x = \frac{\sqrt{345} + 3}{4} \approx 5,393543905$$

$$x=\frac{3-\sqrt{345}}{4}\approx -3,893543905$$

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.

$$2x^{2}-3x+8=50$$

Tenglamaning ikkala tarafidan 50 ni ayirish.

$$2x^{2}-3x+8-50=50-50$$

O‘zidan 50 ayirilsa 0 qoladi.

$$2x^{2}-3x+8-50=0$$

8 dan 50 ni ayirish.

$$2x^{2}-3x-42=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -42 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-42\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-42\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-42\right)}}{2\times 2}$$

-8 ni -42 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+336}}{2\times 2}$$

9 ni 336 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{345}}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{345}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{345}}{4}$$

x=\frac{3±\sqrt{345}}{4} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{345} ga qo'shish.

$$x=\frac{\sqrt{345}+3}{4}$$

x=\frac{3±\sqrt{345}}{4} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{345} ni ayirish.

$$x=\frac{3-\sqrt{345}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{345}+3}{4} x=\frac{3-\sqrt{345}}{4}$$

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.

$$2x^{2}-3x+8=50$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$2x^{2}-3x+8-8=50-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$2x^{2}-3x=50-8$$

50 dan 8 ni ayirish.

$$2x^{2}-3x=42$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-3x}{2}=\frac{42}{2}$$

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

$$x^{2}-\frac{3}{2}x=\frac{42}{2}$$

42 ni 2 ga bo'lish.

$$x^{2}-\frac{3}{2}x=21$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=21+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=21+\frac{9}{16}$$

21 ni \frac{9}{16} ga qo'shish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{345}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{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.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{345}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{345}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{\sqrt{345}}{4} x-\frac{3}{4}=-\frac{\sqrt{345}}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{345}+3}{4} x=\frac{3-\sqrt{345}}{4}$$