30=-8x-49x^2 eching

$$30=-8x-49 { x }^{ 2 }$$

x uchun yechish (complex solution)

$$x=\frac{-4+\sqrt{1454}i}{49}\approx -0,081632653+0,778190856i$$

$$x=\frac{-\sqrt{1454}i-4}{49}\approx -0,081632653-0,778190856i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-8x-49x^{2}=30$$

Ikkala tarafdan 30 ni ayirish.

$$-8x-49x^{2}-30=0$$

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.

$$-49x^{2}-8x-30=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, -8 ni b va -30 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-49\right)\left(-30\right)}}{2\left(-49\right)}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-49\right)\left(-30\right)}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+196\left(-30\right)}}{2\left(-49\right)}$$

196 ni -30 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-5880}}{2\left(-49\right)}$$

64 ni -5880 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{-5816}}{2\left(-49\right)}$$

-5816 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±2\sqrt{1454}i}{2\left(-49\right)}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±2\sqrt{1454}i}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{8±2\sqrt{1454}i}{-98}$$

x=\frac{8±2\sqrt{1454}i}{-98} tenglamasini yeching, bunda ± musbat. 8 ni 2i\sqrt{1454} ga qo'shish.

$$x=\frac{8+2\sqrt{1454}i}{-98}$$

8+2i\sqrt{1454} ni -98 ga bo'lish.

$$x=\frac{-\sqrt{1454}i-4}{49}$$

x=\frac{8±2\sqrt{1454}i}{-98} tenglamasini yeching, bunda ± manfiy. 8 dan 2i\sqrt{1454} ni ayirish.

$$x=\frac{-2\sqrt{1454}i+8}{-98}$$

8-2i\sqrt{1454} ni -98 ga bo'lish.

$$x=\frac{-4+\sqrt{1454}i}{49}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{1454}i-4}{49} x=\frac{-4+\sqrt{1454}i}{49}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-8x-49x^{2}=30$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-49x^{2}-8x=30$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}-8x}{-49}=\frac{30}{-49}$$

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

$$x^{2}+\left(-\frac{8}{-49}\right)x=\frac{30}{-49}$$

-8 ni -49 ga bo'lish.

$$x^{2}+\frac{8}{49}x=\frac{30}{-49}$$

30 ni -49 ga bo'lish.

$$x^{2}+\frac{8}{49}x=-\frac{30}{49}$$

\frac{8}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{4}{49} olish uchun. Keyin, \frac{4}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{8}{49}x+\left(\frac{4}{49}\right)^{2}=-\frac{30}{49}+\left(\frac{4}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{4}{49} kvadratini chiqarish.

$$x^{2}+\frac{8}{49}x+\frac{16}{2401}=-\frac{30}{49}+\frac{16}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{30}{49} ni \frac{16}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{49}x+\frac{16}{2401}=-\frac{1454}{2401}$$

x^{2}+\frac{8}{49}x+\frac{16}{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.

$$\left(x+\frac{4}{49}\right)^{2}=-\frac{1454}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{49}\right)^{2}}=\sqrt{-\frac{1454}{2401}}$$

Qisqartirish.

$$x+\frac{4}{49}=\frac{\sqrt{1454}i}{49} x+\frac{4}{49}=-\frac{\sqrt{1454}i}{49}$$

Tenglamaning ikkala tarafidan \frac{4}{49} ni ayirish.

$$x=\frac{-4+\sqrt{1454}i}{49} x=\frac{-\sqrt{1454}i-4}{49}$$