(40-x)(20+2x)=1500 eching

$$(40-x)(20+2x)=1500$$

x uchun yechish (complex solution)

$$x=15+5\sqrt{5}i\approx 15+11,180339887i$$

$$x=-5\sqrt{5}i+15\approx 15-11,180339887i$$

Kvadrat tenglamalaridan foydalanish qoidalari

40-x ga 20+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$800+60x-2x^{2}=1500$$

Ikkala tarafdan 1500 ni ayirish.

$$800+60x-2x^{2}-1500=0$$

-700 olish uchun 800 dan 1500 ni ayirish.

$$-700+60x-2x^{2}=0$$

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.

$$-2x^{2}+60x-700=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 60 ni b va -700 ni c bilan almashtiring.

$$x=\frac{-60±\sqrt{60^{2}-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

60 kvadratini chiqarish.

$$x=\frac{-60±\sqrt{3600-4\left(-2\right)\left(-700\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-60±\sqrt{3600+8\left(-700\right)}}{2\left(-2\right)}$$

8 ni -700 marotabaga ko'paytirish.

$$x=\frac{-60±\sqrt{3600-5600}}{2\left(-2\right)}$$

3600 ni -5600 ga qo'shish.

$$x=\frac{-60±\sqrt{-2000}}{2\left(-2\right)}$$

-2000 ning kvadrat ildizini chiqarish.

$$x=\frac{-60±20\sqrt{5}i}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-60±20\sqrt{5}i}{-4}$$

x=\frac{-60±20\sqrt{5}i}{-4} tenglamasini yeching, bunda ± musbat. -60 ni 20i\sqrt{5} ga qo'shish.

$$x=\frac{-60+20\sqrt{5}i}{-4}$$

-60+20i\sqrt{5} ni -4 ga bo'lish.

$$x=-5\sqrt{5}i+15$$

x=\frac{-60±20\sqrt{5}i}{-4} tenglamasini yeching, bunda ± manfiy. -60 dan 20i\sqrt{5} ni ayirish.

$$x=\frac{-20\sqrt{5}i-60}{-4}$$

-60-20i\sqrt{5} ni -4 ga bo'lish.

$$x=15+5\sqrt{5}i$$

Tenglama yechildi.

$$x=-5\sqrt{5}i+15 x=15+5\sqrt{5}i$$

Kvadratni yakunlash bo'yicha qadamlar

40-x ga 20+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$800+60x-2x^{2}=1500$$

Ikkala tarafdan 800 ni ayirish.

$$60x-2x^{2}=1500-800$$

700 olish uchun 1500 dan 800 ni ayirish.

$$60x-2x^{2}=700$$

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

$$-2x^{2}+60x=700$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+60x}{-2}=\frac{700}{-2}$$

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

$$x^{2}+\frac{60}{-2}x=\frac{700}{-2}$$

60 ni -2 ga bo'lish.

$$x^{2}-30x=\frac{700}{-2}$$

700 ni -2 ga bo'lish.

$$x^{2}-30x=-350$$

-30 ni bo‘lish, x shartining koeffitsienti, 2 ga -15 olish uchun. Keyin, -15 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-30x+\left(-15\right)^{2}=-350+\left(-15\right)^{2}$$

-15 kvadratini chiqarish.

$$x^{2}-30x+225=-350+225$$

-350 ni 225 ga qo'shish.

$$x^{2}-30x+225=-125$$

x^{2}-30x+225 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-15\right)^{2}=-125$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-15\right)^{2}}=\sqrt{-125}$$

Qisqartirish.

$$x-15=5\sqrt{5}i x-15=-5\sqrt{5}i$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$x=15+5\sqrt{5}i x=-5\sqrt{5}i+15$$