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

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

x uchun yechish

$$x = \frac{\sqrt{589} + 7}{6} \approx 5,2115537$$

$$x=\frac{7-\sqrt{589}}{6}\approx -2,878220367$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x=\left(3x-15\right)\left(x+3\right)$$

3x-15 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x=3x^{2}-6x-45$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-6x-45$$

6x ni ikki tarafga qo’shing.

$$x-3x^{2}+6x=-45$$

7x ni olish uchun x va 6x ni birlashtirish.

$$7x-3x^{2}=-45$$

45 ni ikki tarafga qo’shing.

$$7x-3x^{2}+45=0$$

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.

$$-3x^{2}+7x+45=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 7 ni b va 45 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+12\times 45}}{2\left(-3\right)}$$

12 ni 45 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+540}}{2\left(-3\right)}$$

49 ni 540 ga qo'shish.

$$x=\frac{-7±\sqrt{589}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{589}}{-6}$$

x=\frac{-7±\sqrt{589}}{-6} tenglamasini yeching, bunda ± musbat. -7 ni \sqrt{589} ga qo'shish.

$$x=\frac{\sqrt{589}-7}{-6}$$

-7+\sqrt{589} ni -6 ga bo'lish.

$$x=\frac{7-\sqrt{589}}{6}$$

x=\frac{-7±\sqrt{589}}{-6} tenglamasini yeching, bunda ± manfiy. -7 dan \sqrt{589} ni ayirish.

$$x=\frac{-\sqrt{589}-7}{-6}$$

-7-\sqrt{589} ni -6 ga bo'lish.

$$x=\frac{\sqrt{589}+7}{6}$$

Tenglama yechildi.

$$x=\frac{7-\sqrt{589}}{6} x=\frac{\sqrt{589}+7}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x=\left(3x-15\right)\left(x+3\right)$$

3x-15 ga x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x=3x^{2}-6x-45$$

Ikkala tarafdan 3x^{2} ni ayirish.

$$x-3x^{2}=-6x-45$$

6x ni ikki tarafga qo’shing.

$$x-3x^{2}+6x=-45$$

7x ni olish uchun x va 6x ni birlashtirish.

$$7x-3x^{2}=-45$$

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

$$-3x^{2}+7x=-45$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+7x}{-3}=-\frac{45}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{-3}x=-\frac{45}{-3}$$

7 ni -3 ga bo'lish.

$$x^{2}-\frac{7}{3}x=-\frac{45}{-3}$$

-45 ni -3 ga bo'lish.

$$x^{2}-\frac{7}{3}x=15$$

-\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{6} olish uchun. Keyin, -\frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=15+\left(-\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{6} kvadratini chiqarish.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=15+\frac{49}{36}$$

15 ni \frac{49}{36} ga qo'shish.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{589}{36}$$

x^{2}-\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{589}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{589}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{\sqrt{589}}{6} x-\frac{7}{6}=-\frac{\sqrt{589}}{6}$$

\frac{7}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{589}+7}{6} x=\frac{7-\sqrt{589}}{6}$$