-3(2x-1)+(x+1)(x-1)-5(x+2)=1 eching

$$- 3 ( 2 x - 1 ) + ( x + 1 ) ( x - 1 ) - 5 ( x + 2 ) = 1$$

x uchun yechish

$$x = \frac{\sqrt{157} + 11}{2} \approx 11,764982043$$

$$x=\frac{11-\sqrt{157}}{2}\approx -0,764982043$$

Kvadrat tenglamalaridan foydalanish qoidalari

-3 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6x+3+\left(x+1\right)\left(x-1\right)-5\left(x+2\right)=1$$

Hisoblang: \left(x+1\right)\left(x-1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$-6x+3+x^{2}-1-5\left(x+2\right)=1$$

2 olish uchun 3 dan 1 ni ayirish.

$$-6x+2+x^{2}-5\left(x+2\right)=1$$

-5 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6x+2+x^{2}-5x-10=1$$

-11x ni olish uchun -6x va -5x ni birlashtirish.

$$-11x+2+x^{2}-10=1$$

-8 olish uchun 2 dan 10 ni ayirish.

$$-11x-8+x^{2}=1$$

Ikkala tarafdan 1 ni ayirish.

$$-11x-8+x^{2}-1=0$$

-9 olish uchun -8 dan 1 ni ayirish.

$$-11x-9+x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$x^{2}-11x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -11 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-9\right)}}{2}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-9\right)}}{2}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+36}}{2}$$

121 ni 36 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{157}}{2}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{157}}{2}$$

x=\frac{11±\sqrt{157}}{2} tenglamasini yeching, bunda ± musbat. 11 ni \sqrt{157} ga qo'shish.

$$x=\frac{\sqrt{157}+11}{2}$$

x=\frac{11±\sqrt{157}}{2} tenglamasini yeching, bunda ± manfiy. 11 dan \sqrt{157} ni ayirish.

$$x=\frac{11-\sqrt{157}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{157}+11}{2} x=\frac{11-\sqrt{157}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

-3 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6x+3+\left(x+1\right)\left(x-1\right)-5\left(x+2\right)=1$$

Hisoblang: \left(x+1\right)\left(x-1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$-6x+3+x^{2}-1-5\left(x+2\right)=1$$

2 olish uchun 3 dan 1 ni ayirish.

$$-6x+2+x^{2}-5\left(x+2\right)=1$$

-5 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-6x+2+x^{2}-5x-10=1$$

-11x ni olish uchun -6x va -5x ni birlashtirish.

$$-11x+2+x^{2}-10=1$$

-8 olish uchun 2 dan 10 ni ayirish.

$$-11x-8+x^{2}=1$$

8 ni ikki tarafga qo’shing.

$$-11x+x^{2}=1+8$$

9 olish uchun 1 va 8'ni qo'shing.

$$-11x+x^{2}=9$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$x^{2}-11x=9$$

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

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=9+\left(-\frac{11}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{2} kvadratini chiqarish.

$$x^{2}-11x+\frac{121}{4}=9+\frac{121}{4}$$

9 ni \frac{121}{4} ga qo'shish.

$$x^{2}-11x+\frac{121}{4}=\frac{157}{4}$$

x^{2}-11x+\frac{121}{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.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{157}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{157}{4}}$$

Qisqartirish.

$$x-\frac{11}{2}=\frac{\sqrt{157}}{2} x-\frac{11}{2}=-\frac{\sqrt{157}}{2}$$

\frac{11}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{157}+11}{2} x=\frac{11-\sqrt{157}}{2}$$