-49x^2+9x+22=0 eching

$$-49 { x }^{ 2 } +9x+22=0$$

x uchun yechish

$$x=\frac{\sqrt{4393}+9}{98}\approx 0,768160309$$

$$x=\frac{9-\sqrt{4393}}{98}\approx -0,58448684$$

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.

$$-49x^{2}+9x+22=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 9 ni b va 22 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-49\right)\times 22}}{2\left(-49\right)}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-49\right)\times 22}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+196\times 22}}{2\left(-49\right)}$$

196 ni 22 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+4312}}{2\left(-49\right)}$$

81 ni 4312 ga qo'shish.

$$x=\frac{-9±\sqrt{4393}}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{4393}}{-98}$$

x=\frac{-9±\sqrt{4393}}{-98} tenglamasini yeching, bunda ± musbat. -9 ni \sqrt{4393} ga qo'shish.

$$x=\frac{\sqrt{4393}-9}{-98}$$

-9+\sqrt{4393} ni -98 ga bo'lish.

$$x=\frac{9-\sqrt{4393}}{98}$$

x=\frac{-9±\sqrt{4393}}{-98} tenglamasini yeching, bunda ± manfiy. -9 dan \sqrt{4393} ni ayirish.

$$x=\frac{-\sqrt{4393}-9}{-98}$$

-9-\sqrt{4393} ni -98 ga bo'lish.

$$x=\frac{\sqrt{4393}+9}{98}$$

Tenglama yechildi.

$$x=\frac{9-\sqrt{4393}}{98} x=\frac{\sqrt{4393}+9}{98}$$

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.

$$-49x^{2}+9x+22=0$$

Tenglamaning ikkala tarafidan 22 ni ayirish.

$$-49x^{2}+9x+22-22=-22$$

O‘zidan 22 ayirilsa 0 qoladi.

$$-49x^{2}+9x=-22$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}+9x}{-49}=-\frac{22}{-49}$$

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

$$x^{2}+\frac{9}{-49}x=-\frac{22}{-49}$$

9 ni -49 ga bo'lish.

$$x^{2}-\frac{9}{49}x=-\frac{22}{-49}$$

-22 ni -49 ga bo'lish.

$$x^{2}-\frac{9}{49}x=\frac{22}{49}$$

-\frac{9}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{98} olish uchun. Keyin, -\frac{9}{98} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{49}x+\left(-\frac{9}{98}\right)^{2}=\frac{22}{49}+\left(-\frac{9}{98}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{98} kvadratini chiqarish.

$$x^{2}-\frac{9}{49}x+\frac{81}{9604}=\frac{22}{49}+\frac{81}{9604}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{22}{49} ni \frac{81}{9604} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{49}x+\frac{81}{9604}=\frac{4393}{9604}$$

x^{2}-\frac{9}{49}x+\frac{81}{9604} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{9}{98}\right)^{2}=\frac{4393}{9604}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{98}\right)^{2}}=\sqrt{\frac{4393}{9604}}$$

Qisqartirish.

$$x-\frac{9}{98}=\frac{\sqrt{4393}}{98} x-\frac{9}{98}=-\frac{\sqrt{4393}}{98}$$

\frac{9}{98} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{4393}+9}{98} x=\frac{9-\sqrt{4393}}{98}$$