(2x+3)(x+4)=1 eching

$$(2x+3)(x+4)=1$$

x uchun yechish

$$x=\frac{\sqrt{33}-11}{4}\approx -1,313859338$$

$$x=\frac{-\sqrt{33}-11}{4}\approx -4,186140662$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x+3 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+11x+12=1$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}+11x+12-1=0$$

11 olish uchun 12 dan 1 ni ayirish.

$$2x^{2}+11x+11=0$$

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 11 ni c bilan almashtiring.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\times 11}}{2\times 2}$$

11 kvadratini chiqarish.

$$x=\frac{-11±\sqrt{121-4\times 2\times 11}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121-8\times 11}}{2\times 2}$$

-8 ni 11 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121-88}}{2\times 2}$$

121 ni -88 ga qo'shish.

$$x=\frac{-11±\sqrt{33}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{33}}{4}$$

x=\frac{-11±\sqrt{33}}{4} tenglamasini yeching, bunda ± musbat. -11 ni \sqrt{33} ga qo'shish.

$$x=\frac{\sqrt{33}-11}{4}$$

x=\frac{-11±\sqrt{33}}{4} tenglamasini yeching, bunda ± manfiy. -11 dan \sqrt{33} ni ayirish.

$$x=\frac{-\sqrt{33}-11}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{33}-11}{4} x=\frac{-\sqrt{33}-11}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

2x+3 ga x+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+11x+12=1$$

Ikkala tarafdan 12 ni ayirish.

$$2x^{2}+11x=1-12$$

-11 olish uchun 1 dan 12 ni ayirish.

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

Ikki tarafini 2 ga bo‘ling.

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

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

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

\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.

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

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

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{11}{2} ni \frac{121}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{33}{16}$$

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.

$$\left(x+\frac{11}{4}\right)^{2}=\frac{33}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{33}{16}}$$

Qisqartirish.

$$x+\frac{11}{4}=\frac{\sqrt{33}}{4} x+\frac{11}{4}=-\frac{\sqrt{33}}{4}$$

Tenglamaning ikkala tarafidan \frac{11}{4} ni ayirish.

$$x=\frac{\sqrt{33}-11}{4} x=\frac{-\sqrt{33}-11}{4}$$