450pi=2pix(x+15) eching

$$450 \pi =2 \pi x(x+15)$$

x uchun yechish

$$x = \frac{15 \sqrt{5} - 15}{2} \approx 9,270509831$$

$$x=\frac{-15\sqrt{5}-15}{2}\approx -24,270509831$$

Kvadrat tenglamalaridan foydalanish qoidalari

\pi ni ikki tarafidan bekor qilish.

$$450=2x\left(x+15\right)$$

2x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$450=2x^{2}+30x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2x^{2}+30x=450$$

Ikkala tarafdan 450 ni ayirish.

$$2x^{2}+30x-450=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 30 ni b va -450 ni c bilan almashtiring.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2\left(-450\right)}}{2\times 2}$$

30 kvadratini chiqarish.

$$x=\frac{-30±\sqrt{900-4\times 2\left(-450\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900-8\left(-450\right)}}{2\times 2}$$

-8 ni -450 marotabaga ko'paytirish.

$$x=\frac{-30±\sqrt{900+3600}}{2\times 2}$$

900 ni 3600 ga qo'shish.

$$x=\frac{-30±\sqrt{4500}}{2\times 2}$$

4500 ning kvadrat ildizini chiqarish.

$$x=\frac{-30±30\sqrt{5}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-30±30\sqrt{5}}{4}$$

x=\frac{-30±30\sqrt{5}}{4} tenglamasini yeching, bunda ± musbat. -30 ni 30\sqrt{5} ga qo'shish.

$$x=\frac{30\sqrt{5}-30}{4}$$

-30+30\sqrt{5} ni 4 ga bo'lish.

$$x=\frac{15\sqrt{5}-15}{2}$$

x=\frac{-30±30\sqrt{5}}{4} tenglamasini yeching, bunda ± manfiy. -30 dan 30\sqrt{5} ni ayirish.

$$x=\frac{-30\sqrt{5}-30}{4}$$

-30-30\sqrt{5} ni 4 ga bo'lish.

$$x=\frac{-15\sqrt{5}-15}{2}$$

Tenglama yechildi.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\pi ni ikki tarafidan bekor qilish.

$$450=2x\left(x+15\right)$$

2x ga x+15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$450=2x^{2}+30x$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$2x^{2}+30x=450$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+30x}{2}=\frac{450}{2}$$

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

$$x^{2}+\frac{30}{2}x=\frac{450}{2}$$

30 ni 2 ga bo'lish.

$$x^{2}+15x=\frac{450}{2}$$

450 ni 2 ga bo'lish.

$$x^{2}+15x=225$$

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

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=225+\left(\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{2} kvadratini chiqarish.

$$x^{2}+15x+\frac{225}{4}=225+\frac{225}{4}$$

225 ni \frac{225}{4} ga qo'shish.

$$x^{2}+15x+\frac{225}{4}=\frac{1125}{4}$$

x^{2}+15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{1125}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1125}{4}}$$

Qisqartirish.

$$x+\frac{15}{2}=\frac{15\sqrt{5}}{2} x+\frac{15}{2}=-\frac{15\sqrt{5}}{2}$$

Tenglamaning ikkala tarafidan \frac{15}{2} ni ayirish.

$$x=\frac{15\sqrt{5}-15}{2} x=\frac{-15\sqrt{5}-15}{2}$$