(16-x)(9-x)=112 eching

$$( 16 - x ) ( 9 - x ) = 112$$

x uchun yechish

$$x = \frac{25 - \sqrt{497}}{2} \approx 1,353251595$$

$$x = \frac{\sqrt{497} + 25}{2} \approx 23,646748405$$

Kvadrat tenglamalaridan foydalanish qoidalari

16-x ga 9-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$144-25x+x^{2}=112$$

Ikkala tarafdan 112 ni ayirish.

$$144-25x+x^{2}-112=0$$

32 olish uchun 144 dan 112 ni ayirish.

$$32-25x+x^{2}=0$$

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.

$$x^{2}-25x+32=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -25 ni b va 32 ni c bilan almashtiring.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 32}}{2}$$

-25 kvadratini chiqarish.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 32}}{2}$$

-4 ni 32 marotabaga ko'paytirish.

$$x=\frac{-\left(-25\right)±\sqrt{625-128}}{2}$$

625 ni -128 ga qo'shish.

$$x=\frac{-\left(-25\right)±\sqrt{497}}{2}$$

-25 ning teskarisi 25 ga teng.

$$x=\frac{25±\sqrt{497}}{2}$$

x=\frac{25±\sqrt{497}}{2} tenglamasini yeching, bunda ± musbat. 25 ni \sqrt{497} ga qo'shish.

$$x=\frac{\sqrt{497}+25}{2}$$

x=\frac{25±\sqrt{497}}{2} tenglamasini yeching, bunda ± manfiy. 25 dan \sqrt{497} ni ayirish.

$$x=\frac{25-\sqrt{497}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

16-x ga 9-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$144-25x+x^{2}=112$$

Ikkala tarafdan 144 ni ayirish.

$$-25x+x^{2}=112-144$$

-32 olish uchun 112 dan 144 ni ayirish.

$$-25x+x^{2}=-32$$

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

$$x^{2}-25x=-32$$

-25 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{25}{2} olish uchun. Keyin, -\frac{25}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-32+\left(-\frac{25}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{25}{2} kvadratini chiqarish.

$$x^{2}-25x+\frac{625}{4}=-32+\frac{625}{4}$$

-32 ni \frac{625}{4} ga qo'shish.

$$x^{2}-25x+\frac{625}{4}=\frac{497}{4}$$

x^{2}-25x+\frac{625}{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.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{497}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{497}{4}}$$

Qisqartirish.

$$x-\frac{25}{2}=\frac{\sqrt{497}}{2} x-\frac{25}{2}=-\frac{\sqrt{497}}{2}$$

\frac{25}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{497}+25}{2} x=\frac{25-\sqrt{497}}{2}$$