-49x^2+307x+248=0 eching

$$-49 { x }^{ 2 } +307x+248=0$$

x uchun yechish

$$x = \frac{3 \sqrt{15873} + 307}{98} \approx 6,98943147$$

$$x=\frac{307-3\sqrt{15873}}{98}\approx -0,724125347$$

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.

$$-49x^{2}+307x+248=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 307 ni b va 248 ni c bilan almashtiring.

$$x=\frac{-307±\sqrt{307^{2}-4\left(-49\right)\times 248}}{2\left(-49\right)}$$

307 kvadratini chiqarish.

$$x=\frac{-307±\sqrt{94249-4\left(-49\right)\times 248}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-307±\sqrt{94249+196\times 248}}{2\left(-49\right)}$$

196 ni 248 marotabaga ko'paytirish.

$$x=\frac{-307±\sqrt{94249+48608}}{2\left(-49\right)}$$

94249 ni 48608 ga qo'shish.

$$x=\frac{-307±\sqrt{142857}}{2\left(-49\right)}$$

142857 ning kvadrat ildizini chiqarish.

$$x=\frac{-307±3\sqrt{15873}}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{-307±3\sqrt{15873}}{-98}$$

x=\frac{-307±3\sqrt{15873}}{-98} tenglamasini yeching, bunda ± musbat. -307 ni 3\sqrt{15873} ga qo'shish.

$$x=\frac{3\sqrt{15873}-307}{-98}$$

-307+3\sqrt{15873} ni -98 ga bo'lish.

$$x=\frac{307-3\sqrt{15873}}{98}$$

x=\frac{-307±3\sqrt{15873}}{-98} tenglamasini yeching, bunda ± manfiy. -307 dan 3\sqrt{15873} ni ayirish.

$$x=\frac{-3\sqrt{15873}-307}{-98}$$

-307-3\sqrt{15873} ni -98 ga bo'lish.

$$x=\frac{3\sqrt{15873}+307}{98}$$

Tenglama yechildi.

$$x=\frac{307-3\sqrt{15873}}{98} x=\frac{3\sqrt{15873}+307}{98}$$

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.

$$-49x^{2}+307x+248=0$$

Tenglamaning ikkala tarafidan 248 ni ayirish.

$$-49x^{2}+307x+248-248=-248$$

O‘zidan 248 ayirilsa 0 qoladi.

$$-49x^{2}+307x=-248$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}+307x}{-49}=-\frac{248}{-49}$$

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

$$x^{2}+\frac{307}{-49}x=-\frac{248}{-49}$$

307 ni -49 ga bo'lish.

$$x^{2}-\frac{307}{49}x=-\frac{248}{-49}$$

-248 ni -49 ga bo'lish.

$$x^{2}-\frac{307}{49}x=\frac{248}{49}$$

-\frac{307}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{307}{98} olish uchun. Keyin, -\frac{307}{98} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{307}{49}x+\left(-\frac{307}{98}\right)^{2}=\frac{248}{49}+\left(-\frac{307}{98}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{307}{98} kvadratini chiqarish.

$$x^{2}-\frac{307}{49}x+\frac{94249}{9604}=\frac{248}{49}+\frac{94249}{9604}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{248}{49} ni \frac{94249}{9604} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{307}{49}x+\frac{94249}{9604}=\frac{142857}{9604}$$

x^{2}-\frac{307}{49}x+\frac{94249}{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.

$$\left(x-\frac{307}{98}\right)^{2}=\frac{142857}{9604}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{307}{98}\right)^{2}}=\sqrt{\frac{142857}{9604}}$$

Qisqartirish.

$$x-\frac{307}{98}=\frac{3\sqrt{15873}}{98} x-\frac{307}{98}=-\frac{3\sqrt{15873}}{98}$$

\frac{307}{98} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{15873}+307}{98} x=\frac{307-3\sqrt{15873}}{98}$$