3000=(125+x)(45-x) eching

$$3000=(125+x)(45-x)$$

x uchun yechish

$$x=-105$$

$$x=25$$

Kvadrat tenglamalaridan foydalanish qoidalari

125+x ga 45-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3000=5625-80x-x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$5625-80x-x^{2}=3000$$

Ikkala tarafdan 3000 ni ayirish.

$$5625-80x-x^{2}-3000=0$$

2625 olish uchun 5625 dan 3000 ni ayirish.

$$2625-80x-x^{2}=0$$

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.

$$-x^{2}-80x+2625=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -80 ni b va 2625 ni c bilan almashtiring.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\left(-1\right)\times 2625}}{2\left(-1\right)}$$

-80 kvadratini chiqarish.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\left(-1\right)\times 2625}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400+4\times 2625}}{2\left(-1\right)}$$

4 ni 2625 marotabaga ko'paytirish.

$$x=\frac{-\left(-80\right)±\sqrt{6400+10500}}{2\left(-1\right)}$$

6400 ni 10500 ga qo'shish.

$$x=\frac{-\left(-80\right)±\sqrt{16900}}{2\left(-1\right)}$$

16900 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-80\right)±130}{2\left(-1\right)}$$

-80 ning teskarisi 80 ga teng.

$$x=\frac{80±130}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{80±130}{-2}$$

x=\frac{80±130}{-2} tenglamasini yeching, bunda ± musbat. 80 ni 130 ga qo'shish.

$$x=\frac{210}{-2}$$

210 ni -2 ga bo'lish.

$$x=-105$$

x=\frac{80±130}{-2} tenglamasini yeching, bunda ± manfiy. 80 dan 130 ni ayirish.

$$x=-\frac{50}{-2}$$

-50 ni -2 ga bo'lish.

$$x=25$$

Tenglama yechildi.

$$x=-105 x=25$$

Kvadratni yakunlash bo'yicha qadamlar

125+x ga 45-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3000=5625-80x-x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$5625-80x-x^{2}=3000$$

Ikkala tarafdan 5625 ni ayirish.

$$-80x-x^{2}=3000-5625$$

-2625 olish uchun 3000 dan 5625 ni ayirish.

$$-80x-x^{2}=-2625$$

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

$$-x^{2}-80x=-2625$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-80x}{-1}=-\frac{2625}{-1}$$

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

$$x^{2}+\left(-\frac{80}{-1}\right)x=-\frac{2625}{-1}$$

-80 ni -1 ga bo'lish.

$$x^{2}+80x=-\frac{2625}{-1}$$

-2625 ni -1 ga bo'lish.

$$x^{2}+80x=2625$$

80 ni bo‘lish, x shartining koeffitsienti, 2 ga 40 olish uchun. Keyin, 40 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+80x+40^{2}=2625+40^{2}$$

40 kvadratini chiqarish.

$$x^{2}+80x+1600=2625+1600$$

2625 ni 1600 ga qo'shish.

$$x^{2}+80x+1600=4225$$

x^{2}+80x+1600 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+40\right)^{2}=4225$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{4225}$$

Qisqartirish.

$$x+40=65 x+40=-65$$

Tenglamaning ikkala tarafidan 40 ni ayirish.

$$x=25 x=-105$$