2x^2-5x+625=5825 eching

$$2 { x }^{ 2 } -5x+625=5825$$

x uchun yechish

$$x = \frac{15 \sqrt{185} + 5}{4} \approx 52,255514408$$

$$x=\frac{5-15\sqrt{185}}{4}\approx -49,755514408$$

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.

$$2x^{2}-5x+625=5825$$

Tenglamaning ikkala tarafidan 5825 ni ayirish.

$$2x^{2}-5x+625-5825=5825-5825$$

O‘zidan 5825 ayirilsa 0 qoladi.

$$2x^{2}-5x+625-5825=0$$

625 dan 5825 ni ayirish.

$$2x^{2}-5x-5200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -5 ni b va -5200 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-5200\right)}}{2\times 2}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-5200\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-5200\right)}}{2\times 2}$$

-8 ni -5200 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+41600}}{2\times 2}$$

25 ni 41600 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{41625}}{2\times 2}$$

41625 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±15\sqrt{185}}{2\times 2}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±15\sqrt{185}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{5±15\sqrt{185}}{4}$$

x=\frac{5±15\sqrt{185}}{4} tenglamasini yeching, bunda ± musbat. 5 ni 15\sqrt{185} ga qo'shish.

$$x=\frac{15\sqrt{185}+5}{4}$$

x=\frac{5±15\sqrt{185}}{4} tenglamasini yeching, bunda ± manfiy. 5 dan 15\sqrt{185} ni ayirish.

$$x=\frac{5-15\sqrt{185}}{4}$$

Tenglama yechildi.

$$x=\frac{15\sqrt{185}+5}{4} x=\frac{5-15\sqrt{185}}{4}$$

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.

$$2x^{2}-5x+625=5825$$

Tenglamaning ikkala tarafidan 625 ni ayirish.

$$2x^{2}-5x+625-625=5825-625$$

O‘zidan 625 ayirilsa 0 qoladi.

$$2x^{2}-5x=5825-625$$

5825 dan 625 ni ayirish.

$$2x^{2}-5x=5200$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-5x}{2}=\frac{5200}{2}$$

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

$$x^{2}-\frac{5}{2}x=\frac{5200}{2}$$

5200 ni 2 ga bo'lish.

$$x^{2}-\frac{5}{2}x=2600$$

-\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{4} olish uchun. Keyin, -\frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=2600+\left(-\frac{5}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{4} kvadratini chiqarish.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=2600+\frac{25}{16}$$

2600 ni \frac{25}{16} ga qo'shish.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{41625}{16}$$

x^{2}-\frac{5}{2}x+\frac{25}{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.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{41625}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{41625}{16}}$$

Qisqartirish.

$$x-\frac{5}{4}=\frac{15\sqrt{185}}{4} x-\frac{5}{4}=-\frac{15\sqrt{185}}{4}$$

\frac{5}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15\sqrt{185}+5}{4} x=\frac{5-15\sqrt{185}}{4}$$