10(10+8)=x(3+x) eching

$$10 ( 10 + 8 ) = x ( 3 + x )$$

x uchun yechish

$$x=-15$$

$$x=12$$

Kvadrat tenglamalaridan foydalanish qoidalari

18 olish uchun 10 va 8'ni qo'shing.

$$10\times 18=x\left(3+x\right)$$

180 hosil qilish uchun 10 va 18 ni ko'paytirish.

$$180=x\left(3+x\right)$$

x ga 3+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$180=3x+x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$3x+x^{2}=180$$

Ikkala tarafdan 180 ni ayirish.

$$3x+x^{2}-180=0$$

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.

$$x^{2}+3x-180=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 3 ni b va -180 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-180\right)}}{2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-180\right)}}{2}$$

-4 ni -180 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+720}}{2}$$

9 ni 720 ga qo'shish.

$$x=\frac{-3±\sqrt{729}}{2}$$

729 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±27}{2}$$

x=\frac{-3±27}{2} tenglamasini yeching, bunda ± musbat. -3 ni 27 ga qo'shish.

$$x=\frac{24}{2}$$

24 ni 2 ga bo'lish.

$$x=12$$

x=\frac{-3±27}{2} tenglamasini yeching, bunda ± manfiy. -3 dan 27 ni ayirish.

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

-30 ni 2 ga bo'lish.

$$x=-15$$

Tenglama yechildi.

$$x=12 x=-15$$

Kvadratni yakunlash bo'yicha qadamlar

18 olish uchun 10 va 8'ni qo'shing.

$$10\times 18=x\left(3+x\right)$$

180 hosil qilish uchun 10 va 18 ni ko'paytirish.

$$180=x\left(3+x\right)$$

x ga 3+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$180=3x+x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$3x+x^{2}=180$$

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

$$x^{2}+3x=180$$

3 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{2} olish uchun. Keyin, \frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=180+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=180+\frac{9}{4}$$

180 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=\frac{729}{4}$$

x^{2}+3x+\frac{9}{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.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{729}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{27}{2} x+\frac{3}{2}=-\frac{27}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=12 x=-15$$