(100+2x)(60+2x)=200*60 eching

$$( 100 + 2 x ) ( 60 + 2 x ) = 200 \times 60$$

x uchun yechish

$$x=10\sqrt{31}-40\approx 15,677643628$$

$$x=-10\sqrt{31}-40\approx -95,677643628$$

Kvadrat tenglamalaridan foydalanish qoidalari

100+2x ga 60+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6000+320x+4x^{2}=200\times 60$$

12000 hosil qilish uchun 200 va 60 ni ko'paytirish.

$$6000+320x+4x^{2}=12000$$

Ikkala tarafdan 12000 ni ayirish.

$$6000+320x+4x^{2}-12000=0$$

-6000 olish uchun 6000 dan 12000 ni ayirish.

$$-6000+320x+4x^{2}=0$$

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.

$$4x^{2}+320x-6000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 320 ni b va -6000 ni c bilan almashtiring.

$$x=\frac{-320±\sqrt{320^{2}-4\times 4\left(-6000\right)}}{2\times 4}$$

320 kvadratini chiqarish.

$$x=\frac{-320±\sqrt{102400-4\times 4\left(-6000\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-320±\sqrt{102400-16\left(-6000\right)}}{2\times 4}$$

-16 ni -6000 marotabaga ko'paytirish.

$$x=\frac{-320±\sqrt{102400+96000}}{2\times 4}$$

102400 ni 96000 ga qo'shish.

$$x=\frac{-320±\sqrt{198400}}{2\times 4}$$

198400 ning kvadrat ildizini chiqarish.

$$x=\frac{-320±80\sqrt{31}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-320±80\sqrt{31}}{8}$$

x=\frac{-320±80\sqrt{31}}{8} tenglamasini yeching, bunda ± musbat. -320 ni 80\sqrt{31} ga qo'shish.

$$x=\frac{80\sqrt{31}-320}{8}$$

-320+80\sqrt{31} ni 8 ga bo'lish.

$$x=10\sqrt{31}-40$$

x=\frac{-320±80\sqrt{31}}{8} tenglamasini yeching, bunda ± manfiy. -320 dan 80\sqrt{31} ni ayirish.

$$x=\frac{-80\sqrt{31}-320}{8}$$

-320-80\sqrt{31} ni 8 ga bo'lish.

$$x=-10\sqrt{31}-40$$

Tenglama yechildi.

$$x=10\sqrt{31}-40 x=-10\sqrt{31}-40$$

Kvadratni yakunlash bo'yicha qadamlar

100+2x ga 60+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$6000+320x+4x^{2}=200\times 60$$

12000 hosil qilish uchun 200 va 60 ni ko'paytirish.

$$6000+320x+4x^{2}=12000$$

Ikkala tarafdan 6000 ni ayirish.

$$320x+4x^{2}=12000-6000$$

6000 olish uchun 12000 dan 6000 ni ayirish.

$$320x+4x^{2}=6000$$

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

$$4x^{2}+320x=6000$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+320x}{4}=\frac{6000}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{320}{4}x=\frac{6000}{4}$$

320 ni 4 ga bo'lish.

$$x^{2}+80x=\frac{6000}{4}$$

6000 ni 4 ga bo'lish.

$$x^{2}+80x=1500$$

80 ni bo‘lish, x shartining koeffitsienti, 2 ga 40 olish uchun. Keyin, 40 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+80x+40^{2}=1500+40^{2}$$

40 kvadratini chiqarish.

$$x^{2}+80x+1600=1500+1600$$

1500 ni 1600 ga qo'shish.

$$x^{2}+80x+1600=3100$$

x^{2}+80x+1600 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+40\right)^{2}=3100$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{3100}$$

Qisqartirish.

$$x+40=10\sqrt{31} x+40=-10\sqrt{31}$$

Tenglamaning ikkala tarafidan 40 ni ayirish.

$$x=10\sqrt{31}-40 x=-10\sqrt{31}-40$$