2x(x+15)=-1 eching

$$2 x ( x + 15 ) = - 1$$

x uchun yechish

$$x=\frac{\sqrt{223}-15}{2}\approx -0,033407738$$

$$x=\frac{-\sqrt{223}-15}{2}\approx -14,966592262$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+30x=-1$$

1 ni ikki tarafga qo’shing.

$$2x^{2}+30x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 30 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2}}{2\times 2}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\times 2}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-8}}{2\times 2}$$

900 ni -8 ga qo'shish.

$$x=\frac{-30±\sqrt{892}}{2\times 2}$$

892 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±2\sqrt{223}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-30±2\sqrt{223}}{4}$$

x=\frac{-30±2\sqrt{223}}{4} tenglamasini yeching, bunda ± musbat. -30 ni 2\sqrt{223} ga qo'shish.

$$x=\frac{2\sqrt{223}-30}{4}$$

-30+2\sqrt{223} ni 4 ga bo'lish.

$$x=\frac{\sqrt{223}-15}{2}$$

x=\frac{-30±2\sqrt{223}}{4} tenglamasini yeching, bunda ± manfiy. -30 dan 2\sqrt{223} ni ayirish.

$$x=\frac{-2\sqrt{223}-30}{4}$$

-30-2\sqrt{223} ni 4 ga bo'lish.

$$x=\frac{-\sqrt{223}-15}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{223}-15}{2} x=\frac{-\sqrt{223}-15}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}+30x=-1$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+30x}{2}=-\frac{1}{2}$$

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

$$x^{2}+\frac{30}{2}x=-\frac{1}{2}$$

30 ni 2 ga bo'lish.

$$x^{2}+15x=-\frac{1}{2}$$

15 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{15}{2} olish uchun. Keyin, \frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=-\frac{1}{2}+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$x^{2}+15x+\frac{225}{4}=-\frac{1}{2}+\frac{225}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{225}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+15x+\frac{225}{4}=\frac{223}{4}$$

x^{2}+15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{223}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{223}{4}}$$

Qisqartirish.

$$x+\frac{15}{2}=\frac{\sqrt{223}}{2} x+\frac{15}{2}=-\frac{\sqrt{223}}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$x=\frac{\sqrt{223}-15}{2} x=\frac{-\sqrt{223}-15}{2}$$