126=(x-10)(35-x) eching

$$126=(x-10)(35-x)$$

x uchun yechish

$$x=17$$

$$x=28$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-10 ga 35-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$126=45x-x^{2}-350$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$45x-x^{2}-350=126$$

Ikkala tarafdan 126 ni ayirish.

$$45x-x^{2}-350-126=0$$

-476 olish uchun -350 dan 126 ni ayirish.

$$45x-x^{2}-476=0$$

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.

$$-x^{2}+45x-476=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 45 ni b va -476 ni c bilan almashtiring.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-1\right)\left(-476\right)}}{2\left(-1\right)}$$

45 kvadratini chiqarish.

$$x=\frac{-45±\sqrt{2025-4\left(-1\right)\left(-476\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-45±\sqrt{2025+4\left(-476\right)}}{2\left(-1\right)}$$

4 ni -476 marotabaga ko'paytirish.

$$x=\frac{-45±\sqrt{2025-1904}}{2\left(-1\right)}$$

2025 ni -1904 ga qo'shish.

$$x=\frac{-45±\sqrt{121}}{2\left(-1\right)}$$

121 ning kvadrat ildizini chiqarish.

$$x=\frac{-45±11}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-45±11}{-2}$$

x=\frac{-45±11}{-2} tenglamasini yeching, bunda ± musbat. -45 ni 11 ga qo'shish.

$$x=-\frac{34}{-2}$$

-34 ni -2 ga bo'lish.

$$x=17$$

x=\frac{-45±11}{-2} tenglamasini yeching, bunda ± manfiy. -45 dan 11 ni ayirish.

$$x=-\frac{56}{-2}$$

-56 ni -2 ga bo'lish.

$$x=28$$

Tenglama yechildi.

$$x=17 x=28$$

Kvadratni yakunlash bo'yicha qadamlar

x-10 ga 35-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$126=45x-x^{2}-350$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$45x-x^{2}-350=126$$

350 ni ikki tarafga qo’shing.

$$45x-x^{2}=126+350$$

476 olish uchun 126 va 350'ni qo'shing.

$$45x-x^{2}=476$$

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

$$-x^{2}+45x=476$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+45x}{-1}=\frac{476}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{45}{-1}x=\frac{476}{-1}$$

45 ni -1 ga bo'lish.

$$x^{2}-45x=\frac{476}{-1}$$

476 ni -1 ga bo'lish.

$$x^{2}-45x=-476$$

-45 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{45}{2} olish uchun. Keyin, -\frac{45}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-45x+\left(-\frac{45}{2}\right)^{2}=-476+\left(-\frac{45}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{45}{2} kvadratini chiqarish.

$$x^{2}-45x+\frac{2025}{4}=-476+\frac{2025}{4}$$

-476 ni \frac{2025}{4} ga qo'shish.

$$x^{2}-45x+\frac{2025}{4}=\frac{121}{4}$$

x^{2}-45x+\frac{2025}{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.

$$\left(x-\frac{45}{2}\right)^{2}=\frac{121}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{45}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Qisqartirish.

$$x-\frac{45}{2}=\frac{11}{2} x-\frac{45}{2}=-\frac{11}{2}$$

\frac{45}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=28 x=17$$