-21x^2+77x=-30+18x eching

$$- 21 x ^ { 2 } + 77 x = - 30 + 18 x$$

x uchun yechish

$$x = \frac{\sqrt{6001} + 59}{42} \approx 3,249193372$$

$$x=\frac{59-\sqrt{6001}}{42}\approx -0,439669563$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan -30 ni ayirish.

$$-21x^{2}+77x-\left(-30\right)=18x$$

-30 ning teskarisi 30 ga teng.

$$-21x^{2}+77x+30=18x$$

Ikkala tarafdan 18x ni ayirish.

$$-21x^{2}+77x+30-18x=0$$

59x ni olish uchun 77x va -18x ni birlashtirish.

$$-21x^{2}+59x+30=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -21 ni a, 59 ni b va 30 ni c bilan almashtiring.

$$x=\frac{-59±\sqrt{59^{2}-4\left(-21\right)\times 30}}{2\left(-21\right)}$$

59 kvadratini chiqarish.

$$x=\frac{-59±\sqrt{3481-4\left(-21\right)\times 30}}{2\left(-21\right)}$$

-4 ni -21 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{3481+84\times 30}}{2\left(-21\right)}$$

84 ni 30 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{3481+2520}}{2\left(-21\right)}$$

3481 ni 2520 ga qo'shish.

$$x=\frac{-59±\sqrt{6001}}{2\left(-21\right)}$$

2 ni -21 marotabaga ko'paytirish.

$$x=\frac{-59±\sqrt{6001}}{-42}$$

x=\frac{-59±\sqrt{6001}}{-42} tenglamasini yeching, bunda ± musbat. -59 ni \sqrt{6001} ga qo'shish.

$$x=\frac{\sqrt{6001}-59}{-42}$$

-59+\sqrt{6001} ni -42 ga bo'lish.

$$x=\frac{59-\sqrt{6001}}{42}$$

x=\frac{-59±\sqrt{6001}}{-42} tenglamasini yeching, bunda ± manfiy. -59 dan \sqrt{6001} ni ayirish.

$$x=\frac{-\sqrt{6001}-59}{-42}$$

-59-\sqrt{6001} ni -42 ga bo'lish.

$$x=\frac{\sqrt{6001}+59}{42}$$

Tenglama yechildi.

$$x=\frac{59-\sqrt{6001}}{42} x=\frac{\sqrt{6001}+59}{42}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 18x ni ayirish.

$$-21x^{2}+77x-18x=-30$$

59x ni olish uchun 77x va -18x ni birlashtirish.

$$-21x^{2}+59x=-30$$

Ikki tarafini -21 ga bo‘ling.

$$\frac{-21x^{2}+59x}{-21}=-\frac{30}{-21}$$

-21 ga bo'lish -21 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{59}{-21}x=-\frac{30}{-21}$$

59 ni -21 ga bo'lish.

$$x^{2}-\frac{59}{21}x=-\frac{30}{-21}$$

\frac{-30}{-21} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{59}{21}x=\frac{10}{7}$$

-\frac{59}{21} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{59}{42} olish uchun. Keyin, -\frac{59}{42} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{59}{21}x+\left(-\frac{59}{42}\right)^{2}=\frac{10}{7}+\left(-\frac{59}{42}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{59}{42} kvadratini chiqarish.

$$x^{2}-\frac{59}{21}x+\frac{3481}{1764}=\frac{10}{7}+\frac{3481}{1764}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10}{7} ni \frac{3481}{1764} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{59}{21}x+\frac{3481}{1764}=\frac{6001}{1764}$$

x^{2}-\frac{59}{21}x+\frac{3481}{1764} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{59}{42}\right)^{2}=\frac{6001}{1764}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{59}{42}\right)^{2}}=\sqrt{\frac{6001}{1764}}$$

Qisqartirish.

$$x-\frac{59}{42}=\frac{\sqrt{6001}}{42} x-\frac{59}{42}=-\frac{\sqrt{6001}}{42}$$

\frac{59}{42} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{6001}+59}{42} x=\frac{59-\sqrt{6001}}{42}$$