(100-2x)(x-30)=200 eching

$$(100-2x)(x-30)=200$$

x uchun yechish

$$x=40$$

Kvadrat tenglamalaridan foydalanish qoidalari

100-2x ga x-30 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$160x-3000-2x^{2}=200$$

Ikkala tarafdan 200 ni ayirish.

$$160x-3000-2x^{2}-200=0$$

-3200 olish uchun -3000 dan 200 ni ayirish.

$$160x-3200-2x^{2}=0$$

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.

$$-2x^{2}+160x-3200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 160 ni b va -3200 ni c bilan almashtiring.

$$x=\frac{-160±\sqrt{160^{2}-4\left(-2\right)\left(-3200\right)}}{2\left(-2\right)}$$

160 kvadratini chiqarish.

$$x=\frac{-160±\sqrt{25600-4\left(-2\right)\left(-3200\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-160±\sqrt{25600+8\left(-3200\right)}}{2\left(-2\right)}$$

8 ni -3200 marotabaga ko'paytirish.

$$x=\frac{-160±\sqrt{25600-25600}}{2\left(-2\right)}$$

25600 ni -25600 ga qo'shish.

$$x=\frac{-160±\sqrt{0}}{2\left(-2\right)}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{160}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=-\frac{160}{-4}$$

-160 ni -4 ga bo'lish.

$$x=40$$

Kvadratni yakunlash bo'yicha qadamlar

100-2x ga x-30 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$160x-3000-2x^{2}=200$$

3000 ni ikki tarafga qo’shing.

$$160x-2x^{2}=200+3000$$

3200 olish uchun 200 va 3000'ni qo'shing.

$$160x-2x^{2}=3200$$

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

$$-2x^{2}+160x=3200$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+160x}{-2}=\frac{3200}{-2}$$

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

$$x^{2}+\frac{160}{-2}x=\frac{3200}{-2}$$

160 ni -2 ga bo'lish.

$$x^{2}-80x=\frac{3200}{-2}$$

3200 ni -2 ga bo'lish.

$$x^{2}-80x=-1600$$

-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.

$$x^{2}-80x+\left(-40\right)^{2}=-1600+\left(-40\right)^{2}$$

-40 kvadratini chiqarish.

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

-1600 ni 1600 ga qo'shish.

$$x^{2}-80x+1600=0$$

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.

$$\left(x-40\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-40=0 x-40=0$$

40 ni tenglamaning ikkala tarafiga qo'shish.

$$x=40 x=40$$

Tenglama yechildi. Yechimlar bir xil.

$$x=40$$