2x^2-90x-3600=0 eching

$$2 { x }^{ 2 } -90x-3600=0$$

x uchun yechish

$$x = \frac{15 \sqrt{41} + 45}{2} \approx 70,523431781$$

$$x=\frac{45-15\sqrt{41}}{2}\approx -25,523431781$$

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.

$$2x^{2}-90x-3600=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -90 ni b va -3600 ni c bilan almashtiring.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 2\left(-3600\right)}}{2\times 2}$$

-90 kvadratini chiqarish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 2\left(-3600\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-8\left(-3600\right)}}{2\times 2}$$

-8 ni -3600 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100+28800}}{2\times 2}$$

8100 ni 28800 ga qo'shish.

$$x=\frac{-\left(-90\right)±\sqrt{36900}}{2\times 2}$$

36900 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-90\right)±30\sqrt{41}}{2\times 2}$$

-90 ning teskarisi 90 ga teng.

$$x=\frac{90±30\sqrt{41}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{90±30\sqrt{41}}{4}$$

x=\frac{90±30\sqrt{41}}{4} tenglamasini yeching, bunda ± musbat. 90 ni 30\sqrt{41} ga qo'shish.

$$x=\frac{30\sqrt{41}+90}{4}$$

90+30\sqrt{41} ni 4 ga bo'lish.

$$x=\frac{15\sqrt{41}+45}{2}$$

x=\frac{90±30\sqrt{41}}{4} tenglamasini yeching, bunda ± manfiy. 90 dan 30\sqrt{41} ni ayirish.

$$x=\frac{90-30\sqrt{41}}{4}$$

90-30\sqrt{41} ni 4 ga bo'lish.

$$x=\frac{45-15\sqrt{41}}{2}$$

Tenglama yechildi.

$$x=\frac{15\sqrt{41}+45}{2} x=\frac{45-15\sqrt{41}}{2}$$

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.

$$2x^{2}-90x-3600=0$$

3600 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-90x-3600-\left(-3600\right)=-\left(-3600\right)$$

O‘zidan -3600 ayirilsa 0 qoladi.

$$2x^{2}-90x=-\left(-3600\right)$$

0 dan -3600 ni ayirish.

$$2x^{2}-90x=3600$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-90x}{2}=\frac{3600}{2}$$

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

$$x^{2}+\left(-\frac{90}{2}\right)x=\frac{3600}{2}$$

-90 ni 2 ga bo'lish.

$$x^{2}-45x=\frac{3600}{2}$$

3600 ni 2 ga bo'lish.

$$x^{2}-45x=1800$$

-45 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{45}{2} olish uchun. Keyin, -\frac{45}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=1800+\left(-\frac{45}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{45}{2} kvadratini chiqarish.

$$x^{2}-45x+\frac{2025}{4}=1800+\frac{2025}{4}$$

1800 ni \frac{2025}{4} ga qo'shish.

$$x^{2}-45x+\frac{2025}{4}=\frac{9225}{4}$$

x^{2}-45x+\frac{2025}{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.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{9225}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{9225}{4}}$$

Qisqartirish.

$$x-\frac{45}{2}=\frac{15\sqrt{41}}{2} x-\frac{45}{2}=-\frac{15\sqrt{41}}{2}$$

\frac{45}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15\sqrt{41}+45}{2} x=\frac{45-15\sqrt{41}}{2}$$