49x^2+30x+25=0 eching

$$49 x ^ { 2 } + 30 x + 25 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-15+10\sqrt{10}i}{49}\approx -0,306122449+0,645362788i$$

$$x=\frac{-10\sqrt{10}i-15}{49}\approx -0,306122449-0,645362788i$$

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.

$$49x^{2}+30x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, 30 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\times 49\times 25}}{2\times 49}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\times 49\times 25}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-196\times 25}}{2\times 49}$$

-196 ni 25 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-4900}}{2\times 49}$$

900 ni -4900 ga qo'shish.

$$x=\frac{-30±\sqrt{-4000}}{2\times 49}$$

-4000 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±20\sqrt{10}i}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{-30±20\sqrt{10}i}{98}$$

x=\frac{-30±20\sqrt{10}i}{98} tenglamasini yeching, bunda ± musbat. -30 ni 20i\sqrt{10} ga qo'shish.

$$x=\frac{-30+20\sqrt{10}i}{98}$$

-30+20i\sqrt{10} ni 98 ga bo'lish.

$$x=\frac{-15+10\sqrt{10}i}{49}$$

x=\frac{-30±20\sqrt{10}i}{98} tenglamasini yeching, bunda ± manfiy. -30 dan 20i\sqrt{10} ni ayirish.

$$x=\frac{-20\sqrt{10}i-30}{98}$$

-30-20i\sqrt{10} ni 98 ga bo'lish.

$$x=\frac{-10\sqrt{10}i-15}{49}$$

Tenglama yechildi.

$$x=\frac{-15+10\sqrt{10}i}{49} x=\frac{-10\sqrt{10}i-15}{49}$$

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.

$$49x^{2}+30x+25=0$$

Tenglamaning ikkala tarafidan 25 ni ayirish.

$$49x^{2}+30x+25-25=-25$$

O‘zidan 25 ayirilsa 0 qoladi.

$$49x^{2}+30x=-25$$

Ikki tarafini 49 ga bo‘ling.

$$\frac{49x^{2}+30x}{49}=-\frac{25}{49}$$

49 ga bo'lish 49 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{30}{49}x=-\frac{25}{49}$$

\frac{30}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{15}{49} olish uchun. Keyin, \frac{15}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{30}{49}x+\left(\frac{15}{49}\right)^{2}=-\frac{25}{49}+\left(\frac{15}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{49} kvadratini chiqarish.

$$x^{2}+\frac{30}{49}x+\frac{225}{2401}=-\frac{25}{49}+\frac{225}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{49} ni \frac{225}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{30}{49}x+\frac{225}{2401}=-\frac{1000}{2401}$$

x^{2}+\frac{30}{49}x+\frac{225}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{15}{49}\right)^{2}=-\frac{1000}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{49}\right)^{2}}=\sqrt{-\frac{1000}{2401}}$$

Qisqartirish.

$$x+\frac{15}{49}=\frac{10\sqrt{10}i}{49} x+\frac{15}{49}=-\frac{10\sqrt{10}i}{49}$$

Tenglamaning ikkala tarafidan \frac{15}{49} ni ayirish.

$$x=\frac{-15+10\sqrt{10}i}{49} x=\frac{-10\sqrt{10}i-15}{49}$$