z^2+3z-30=(2z+5)(z+6) eching

$$z ^ { 2 } + 3 z - 30 = ( 2 z + 5 ) ( z + 6 )$$

z uchun yechish

$$z=-\sqrt{11}i-7\approx -7-3,31662479i$$

$$z=-7+\sqrt{11}i\approx -7+3,31662479i$$

Kvadrat tenglamalaridan foydalanish qoidalari

2z+5 ga z+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Ikkala tarafdan 2z^{2} ni ayirish.

$$z^{2}+3z-30-2z^{2}=17z+30$$

-z^{2} ni olish uchun z^{2} va -2z^{2} ni birlashtirish.

$$-z^{2}+3z-30=17z+30$$

Ikkala tarafdan 17z ni ayirish.

$$-z^{2}+3z-30-17z=30$$

-14z ni olish uchun 3z va -17z ni birlashtirish.

$$-z^{2}-14z-30=30$$

Ikkala tarafdan 30 ni ayirish.

$$-z^{2}-14z-30-30=0$$

-60 olish uchun -30 dan 30 ni ayirish.

$$-z^{2}-14z-60=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -14 ni b va -60 ni c bilan almashtiring.

$$z=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

-14 kvadratini chiqarish.

$$z=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$z=\frac{-\left(-14\right)±\sqrt{196+4\left(-60\right)}}{2\left(-1\right)}$$

4 ni -60 marotabaga ko'paytirish.

$$z=\frac{-\left(-14\right)±\sqrt{196-240}}{2\left(-1\right)}$$

196 ni -240 ga qo'shish.

$$z=\frac{-\left(-14\right)±\sqrt{-44}}{2\left(-1\right)}$$

-44 ning kvadrat ildizini chiqarish.

$$z=\frac{-\left(-14\right)±2\sqrt{11}i}{2\left(-1\right)}$$

-14 ning teskarisi 14 ga teng.

$$z=\frac{14±2\sqrt{11}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$z=\frac{14±2\sqrt{11}i}{-2}$$

z=\frac{14±2\sqrt{11}i}{-2} tenglamasini yeching, bunda ± musbat. 14 ni 2i\sqrt{11} ga qo'shish.

$$z=\frac{14+2\sqrt{11}i}{-2}$$

14+2i\sqrt{11} ni -2 ga bo'lish.

$$z=-\sqrt{11}i-7$$

z=\frac{14±2\sqrt{11}i}{-2} tenglamasini yeching, bunda ± manfiy. 14 dan 2i\sqrt{11} ni ayirish.

$$z=\frac{-2\sqrt{11}i+14}{-2}$$

14-2i\sqrt{11} ni -2 ga bo'lish.

$$z=-7+\sqrt{11}i$$

Tenglama yechildi.

$$z=-\sqrt{11}i-7 z=-7+\sqrt{11}i$$

Kvadratni yakunlash bo'yicha qadamlar

2z+5 ga z+6 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Ikkala tarafdan 2z^{2} ni ayirish.

$$z^{2}+3z-30-2z^{2}=17z+30$$

-z^{2} ni olish uchun z^{2} va -2z^{2} ni birlashtirish.

$$-z^{2}+3z-30=17z+30$$

Ikkala tarafdan 17z ni ayirish.

$$-z^{2}+3z-30-17z=30$$

-14z ni olish uchun 3z va -17z ni birlashtirish.

$$-z^{2}-14z-30=30$$

30 ni ikki tarafga qo’shing.

$$-z^{2}-14z=30+30$$

60 olish uchun 30 va 30'ni qo'shing.

$$-z^{2}-14z=60$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-z^{2}-14z}{-1}=\frac{60}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$z^{2}+\left(-\frac{14}{-1}\right)z=\frac{60}{-1}$$

-14 ni -1 ga bo'lish.

$$z^{2}+14z=\frac{60}{-1}$$

60 ni -1 ga bo'lish.

$$z^{2}+14z=-60$$

14 ni bo‘lish, x shartining koeffitsienti, 2 ga 7 olish uchun. Keyin, 7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}+14z+7^{2}=-60+7^{2}$$

7 kvadratini chiqarish.

$$z^{2}+14z+49=-60+49$$

-60 ni 49 ga qo'shish.

$$z^{2}+14z+49=-11$$

z^{2}+14z+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z+7\right)^{2}=-11$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z+7\right)^{2}}=\sqrt{-11}$$

Qisqartirish.

$$z+7=\sqrt{11}i z+7=-\sqrt{11}i$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$z=-7+\sqrt{11}i z=-\sqrt{11}i-7$$