6x+3x(3x+1)+9=90 eching

$$6x+3x(3x+1)+9 = 90$$

x uchun yechish

$$x = \frac{\sqrt{37} - 1}{2} \approx 2,541381265$$

$$x=\frac{-\sqrt{37}-1}{2}\approx -3,541381265$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$6x+9x^{2}+3x+9=90$$

9x ni olish uchun 6x va 3x ni birlashtirish.

$$9x+9x^{2}+9=90$$

Ikkala tarafdan 90 ni ayirish.

$$9x+9x^{2}+9-90=0$$

-81 olish uchun 9 dan 90 ni ayirish.

$$9x+9x^{2}-81=0$$

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.

$$9x^{2}+9x-81=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, 9 ni b va -81 ni c bilan almashtiring.

$$x=\frac{-9±\sqrt{9^{2}-4\times 9\left(-81\right)}}{2\times 9}$$

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\times 9\left(-81\right)}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-36\left(-81\right)}}{2\times 9}$$

-36 ni -81 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+2916}}{2\times 9}$$

81 ni 2916 ga qo'shish.

$$x=\frac{-9±\sqrt{2997}}{2\times 9}$$

2997 ning kvadrat ildizini chiqarish.

$$x=\frac{-9±9\sqrt{37}}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{-9±9\sqrt{37}}{18}$$

x=\frac{-9±9\sqrt{37}}{18} tenglamasini yeching, bunda ± musbat. -9 ni 9\sqrt{37} ga qo'shish.

$$x=\frac{9\sqrt{37}-9}{18}$$

-9+9\sqrt{37} ni 18 ga bo'lish.

$$x=\frac{\sqrt{37}-1}{2}$$

x=\frac{-9±9\sqrt{37}}{18} tenglamasini yeching, bunda ± manfiy. -9 dan 9\sqrt{37} ni ayirish.

$$x=\frac{-9\sqrt{37}-9}{18}$$

-9-9\sqrt{37} ni 18 ga bo'lish.

$$x=\frac{-\sqrt{37}-1}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{37}-1}{2} x=\frac{-\sqrt{37}-1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$6x+9x^{2}+3x+9=90$$

9x ni olish uchun 6x va 3x ni birlashtirish.

$$9x+9x^{2}+9=90$$

Ikkala tarafdan 9 ni ayirish.

$$9x+9x^{2}=90-9$$

81 olish uchun 90 dan 9 ni ayirish.

$$9x+9x^{2}=81$$

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

$$9x^{2}+9x=81$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}+9x}{9}=\frac{81}{9}$$

9 ga bo'lish 9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{9}{9}x=\frac{81}{9}$$

9 ni 9 ga bo'lish.

$$x^{2}+x=\frac{81}{9}$$

81 ni 9 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=9+\frac{1}{4}$$

9 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{37}{4}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=\frac{37}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{37}}{2} x+\frac{1}{2}=-\frac{\sqrt{37}}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{\sqrt{37}-1}{2} x=\frac{-\sqrt{37}-1}{2}$$