100+400=1600+x^2-80x eching

$$100+400=1600+ { x }^{ 2 } -80x$$

x uchun yechish

$$x=10\sqrt{5}+40\approx 62,360679775$$

$$x=40-10\sqrt{5}\approx 17,639320225$$

Kvadrat tenglamalaridan foydalanish qoidalari

500 olish uchun 100 va 400'ni qo'shing.

$$500=1600+x^{2}-80x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$1600+x^{2}-80x=500$$

Ikkala tarafdan 500 ni ayirish.

$$1600+x^{2}-80x-500=0$$

1100 olish uchun 1600 dan 500 ni ayirish.

$$1100+x^{2}-80x=0$$

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.

$$x^{2}-80x+1100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -80 ni b va 1100 ni c bilan almashtiring.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 1100}}{2}$$

-80 kvadratini chiqarish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 1100}}{2}$$

-4 ni 1100 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4400}}{2}$$

6400 ni -4400 ga qo'shish.

$$x=\frac{-\left(-80\right)±\sqrt{2000}}{2}$$

2000 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-80\right)±20\sqrt{5}}{2}$$

-80 ning teskarisi 80 ga teng.

$$x=\frac{80±20\sqrt{5}}{2}$$

x=\frac{80±20\sqrt{5}}{2} tenglamasini yeching, bunda ± musbat. 80 ni 20\sqrt{5} ga qo'shish.

$$x=\frac{20\sqrt{5}+80}{2}$$

80+20\sqrt{5} ni 2 ga bo'lish.

$$x=10\sqrt{5}+40$$

x=\frac{80±20\sqrt{5}}{2} tenglamasini yeching, bunda ± manfiy. 80 dan 20\sqrt{5} ni ayirish.

$$x=\frac{80-20\sqrt{5}}{2}$$

80-20\sqrt{5} ni 2 ga bo'lish.

$$x=40-10\sqrt{5}$$

Tenglama yechildi.

$$x=10\sqrt{5}+40 x=40-10\sqrt{5}$$

Kvadratni yakunlash bo'yicha qadamlar

500 olish uchun 100 va 400'ni qo'shing.

$$500=1600+x^{2}-80x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$1600+x^{2}-80x=500$$

Ikkala tarafdan 1600 ni ayirish.

$$x^{2}-80x=500-1600$$

-1100 olish uchun 500 dan 1600 ni ayirish.

$$x^{2}-80x=-1100$$

-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.

$$x^{2}-80x+\left(-40\right)^{2}=-1100+\left(-40\right)^{2}$$

-40 kvadratini chiqarish.

$$x^{2}-80x+1600=-1100+1600$$

-1100 ni 1600 ga qo'shish.

$$x^{2}-80x+1600=500$$

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.

$$\left(x-40\right)^{2}=500$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-40\right)^{2}}=\sqrt{500}$$

Qisqartirish.

$$x-40=10\sqrt{5} x-40=-10\sqrt{5}$$

40 ni tenglamaning ikkala tarafiga qo'shish.

$$x=10\sqrt{5}+40 x=40-10\sqrt{5}$$