2x^2-7x-2=4x+5 eching

$$2 x ^ { 2 } - 7 x - 2 = 4 x + 5$$

x uchun yechish

$$x = \frac{\sqrt{177} + 11}{4} \approx 6,076033674$$

$$x=\frac{11-\sqrt{177}}{4}\approx -0,576033674$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 4x ni ayirish.

$$2x^{2}-7x-2-4x=5$$

-11x ni olish uchun -7x va -4x ni birlashtirish.

$$2x^{2}-11x-2=5$$

Ikkala tarafdan 5 ni ayirish.

$$2x^{2}-11x-2-5=0$$

-7 olish uchun -2 dan 5 ni ayirish.

$$2x^{2}-11x-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -11 ni b va -7 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 2\left(-7\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\left(-7\right)}}{2\times 2}$$

-8 ni -7 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+56}}{2\times 2}$$

121 ni 56 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{177}}{2\times 2}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{177}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{11±\sqrt{177}}{4}$$

x=\frac{11±\sqrt{177}}{4} tenglamasini yeching, bunda ± musbat. 11 ni \sqrt{177} ga qo'shish.

$$x=\frac{\sqrt{177}+11}{4}$$

x=\frac{11±\sqrt{177}}{4} tenglamasini yeching, bunda ± manfiy. 11 dan \sqrt{177} ni ayirish.

$$x=\frac{11-\sqrt{177}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{177}+11}{4} x=\frac{11-\sqrt{177}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 4x ni ayirish.

$$2x^{2}-7x-2-4x=5$$

-11x ni olish uchun -7x va -4x ni birlashtirish.

$$2x^{2}-11x-2=5$$

2 ni ikki tarafga qo’shing.

$$2x^{2}-11x=5+2$$

7 olish uchun 5 va 2'ni qo'shing.

$$2x^{2}-11x=7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-11x}{2}=\frac{7}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{11}{2}x=\frac{7}{2}$$

-\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{4} olish uchun. Keyin, -\frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{4} kvadratini chiqarish.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{7}{2}+\frac{121}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{2} ni \frac{121}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{177}{16}$$

x^{2}-\frac{11}{2}x+\frac{121}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{177}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{177}{16}}$$

Qisqartirish.

$$x-\frac{11}{4}=\frac{\sqrt{177}}{4} x-\frac{11}{4}=-\frac{\sqrt{177}}{4}$$

\frac{11}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{177}+11}{4} x=\frac{11-\sqrt{177}}{4}$$