(2x+3)(x-1)=15 eching

$$(2x+3)(x-1)=15$$

x uchun yechish

$$x = \frac{\sqrt{145} - 1}{4} \approx 2,760398645$$

$$x=\frac{-\sqrt{145}-1}{4}\approx -3,260398645$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x+3 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3=15$$

Ikkala tarafdan 15 ni ayirish.

$$2x^{2}+x-3-15=0$$

-18 olish uchun -3 dan 15 ni ayirish.

$$2x^{2}+x-18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 1 ni b va -18 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-18\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-8\left(-18\right)}}{2\times 2}$$

-8 ni -18 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+144}}{2\times 2}$$

1 ni 144 ga qo'shish.

$$x=\frac{-1±\sqrt{145}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{145}}{4}$$

x=\frac{-1±\sqrt{145}}{4} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{145} ga qo'shish.

$$x=\frac{\sqrt{145}-1}{4}$$

x=\frac{-1±\sqrt{145}}{4} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{145} ni ayirish.

$$x=\frac{-\sqrt{145}-1}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{145}-1}{4} x=\frac{-\sqrt{145}-1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

2x+3 ga x-1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$2x^{2}+x-3=15$$

3 ni ikki tarafga qo’shing.

$$2x^{2}+x=15+3$$

18 olish uchun 15 va 3'ni qo'shing.

$$2x^{2}+x=18$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+x}{2}=\frac{18}{2}$$

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

$$x^{2}+\frac{1}{2}x=\frac{18}{2}$$

18 ni 2 ga bo'lish.

$$x^{2}+\frac{1}{2}x=9$$

\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{4} olish uchun. Keyin, \frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=9+\left(\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{4} kvadratini chiqarish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=9+\frac{1}{16}$$

9 ni \frac{1}{16} ga qo'shish.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{145}{16}$$

x^{2}+\frac{1}{2}x+\frac{1}{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.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{145}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{145}{16}}$$

Qisqartirish.

$$x+\frac{1}{4}=\frac{\sqrt{145}}{4} x+\frac{1}{4}=-\frac{\sqrt{145}}{4}$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$x=\frac{\sqrt{145}-1}{4} x=\frac{-\sqrt{145}-1}{4}$$