18=6x+x(x-13) eching

$$18=6x+x(x-13)$$

x uchun yechish

$$x=-2$$

$$x=9$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga x-13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$18=6x+x^{2}-13x$$

-7x ni olish uchun 6x va -13x ni birlashtirish.

$$18=-7x+x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-7x+x^{2}=18$$

Ikkala tarafdan 18 ni ayirish.

$$-7x+x^{2}-18=0$$

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.

$$x^{2}-7x-18=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -7 ni b va -18 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-18\right)}}{2}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-18\right)}}{2}$$

-4 ni -18 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2}$$

49 ni 72 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±11}{2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±11}{2}$$

x=\frac{7±11}{2} tenglamasini yeching, bunda ± musbat. 7 ni 11 ga qo'shish.

$$x=\frac{18}{2}$$

18 ni 2 ga bo'lish.

$$x=9$$

x=\frac{7±11}{2} tenglamasini yeching, bunda ± manfiy. 7 dan 11 ni ayirish.

$$x=-\frac{4}{2}$$

-4 ni 2 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=9 x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

x ga x-13 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$18=6x+x^{2}-13x$$

-7x ni olish uchun 6x va -13x ni birlashtirish.

$$18=-7x+x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-7x+x^{2}=18$$

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

$$x^{2}-7x=18$$

-7 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{2} olish uchun. Keyin, -\frac{7}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=18+\left(-\frac{7}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=18+\frac{49}{4}$$

18 ni \frac{49}{4} ga qo'shish.

$$x^{2}-7x+\frac{49}{4}=\frac{121}{4}$$

x^{2}-7x+\frac{49}{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.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{121}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{11}{2} x-\frac{7}{2}=-\frac{11}{2}$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=9 x=-2$$