x^2+684x-2025=0 eching

$${ x }^{ 2 } +684x-2025=0$$

x uchun yechish

$$x=9\sqrt{1469}-342\approx 2,947822141$$

$$x=-9\sqrt{1469}-342\approx -686,947822141$$

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.

$$x^{2}+684x-2025=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 684 ni b va -2025 ni c bilan almashtiring.

$$x=\frac{-684±\sqrt{684^{2}-4\left(-2025\right)}}{2}$$

684 kvadratini chiqarish.

$$x=\frac{-684±\sqrt{467856-4\left(-2025\right)}}{2}$$

-4 ni -2025 marotabaga ko'paytirish.

$$x=\frac{-684±\sqrt{467856+8100}}{2}$$

467856 ni 8100 ga qo'shish.

$$x=\frac{-684±\sqrt{475956}}{2}$$

475956 ning kvadrat ildizini chiqarish.

$$x=\frac{-684±18\sqrt{1469}}{2}$$

x=\frac{-684±18\sqrt{1469}}{2} tenglamasini yeching, bunda ± musbat. -684 ni 18\sqrt{1469} ga qo'shish.

$$x=\frac{18\sqrt{1469}-684}{2}$$

-684+18\sqrt{1469} ni 2 ga bo'lish.

$$x=9\sqrt{1469}-342$$

x=\frac{-684±18\sqrt{1469}}{2} tenglamasini yeching, bunda ± manfiy. -684 dan 18\sqrt{1469} ni ayirish.

$$x=\frac{-18\sqrt{1469}-684}{2}$$

-684-18\sqrt{1469} ni 2 ga bo'lish.

$$x=-9\sqrt{1469}-342$$

Tenglama yechildi.

$$x=9\sqrt{1469}-342 x=-9\sqrt{1469}-342$$

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.

$$x^{2}+684x-2025=0$$

2025 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+684x-2025-\left(-2025\right)=-\left(-2025\right)$$

O‘zidan -2025 ayirilsa 0 qoladi.

$$x^{2}+684x=-\left(-2025\right)$$

0 dan -2025 ni ayirish.

$$x^{2}+684x=2025$$

684 ni bo‘lish, x shartining koeffitsienti, 2 ga 342 olish uchun. Keyin, 342 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+684x+342^{2}=2025+342^{2}$$

342 kvadratini chiqarish.

$$x^{2}+684x+116964=2025+116964$$

2025 ni 116964 ga qo'shish.

$$x^{2}+684x+116964=118989$$

x^{2}+684x+116964 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+342\right)^{2}=118989$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+342\right)^{2}}=\sqrt{118989}$$

Qisqartirish.

$$x+342=9\sqrt{1469} x+342=-9\sqrt{1469}$$

Tenglamaning ikkala tarafidan 342 ni ayirish.

$$x=9\sqrt{1469}-342 x=-9\sqrt{1469}-342$$