(15-x)(5x+500)=4250 eching

$$( 15 - x ) ( 5 x + 500 ) = 4250$$

x uchun yechish

$$x = \frac{5 \sqrt{393} - 85}{2} \approx 7,060569004$$

$$x=\frac{-5\sqrt{393}-85}{2}\approx -92,060569004$$

Kvadrat tenglamalaridan foydalanish qoidalari

15-x ga 5x+500 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-425x+7500-5x^{2}=4250$$

Ikkala tarafdan 4250 ni ayirish.

$$-425x+7500-5x^{2}-4250=0$$

3250 olish uchun 7500 dan 4250 ni ayirish.

$$-425x+3250-5x^{2}=0$$

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.

$$-5x^{2}-425x+3250=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, -425 ni b va 3250 ni c bilan almashtiring.

$$x=\frac{-\left(-425\right)±\sqrt{\left(-425\right)^{2}-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

-425 kvadratini chiqarish.

$$x=\frac{-\left(-425\right)±\sqrt{180625-4\left(-5\right)\times 3250}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-425\right)±\sqrt{180625+20\times 3250}}{2\left(-5\right)}$$

20 ni 3250 marotabaga ko'paytirish.

$$x=\frac{-\left(-425\right)±\sqrt{180625+65000}}{2\left(-5\right)}$$

180625 ni 65000 ga qo'shish.

$$x=\frac{-\left(-425\right)±\sqrt{245625}}{2\left(-5\right)}$$

245625 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-425\right)±25\sqrt{393}}{2\left(-5\right)}$$

-425 ning teskarisi 425 ga teng.

$$x=\frac{425±25\sqrt{393}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{425±25\sqrt{393}}{-10}$$

x=\frac{425±25\sqrt{393}}{-10} tenglamasini yeching, bunda ± musbat. 425 ni 25\sqrt{393} ga qo'shish.

$$x=\frac{25\sqrt{393}+425}{-10}$$

425+25\sqrt{393} ni -10 ga bo'lish.

$$x=\frac{-5\sqrt{393}-85}{2}$$

x=\frac{425±25\sqrt{393}}{-10} tenglamasini yeching, bunda ± manfiy. 425 dan 25\sqrt{393} ni ayirish.

$$x=\frac{425-25\sqrt{393}}{-10}$$

425-25\sqrt{393} ni -10 ga bo'lish.

$$x=\frac{5\sqrt{393}-85}{2}$$

Tenglama yechildi.

$$x=\frac{-5\sqrt{393}-85}{2} x=\frac{5\sqrt{393}-85}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

15-x ga 5x+500 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-425x+7500-5x^{2}=4250$$

Ikkala tarafdan 7500 ni ayirish.

$$-425x-5x^{2}=4250-7500$$

-3250 olish uchun 4250 dan 7500 ni ayirish.

$$-425x-5x^{2}=-3250$$

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

$$-5x^{2}-425x=-3250$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}-425x}{-5}=-\frac{3250}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{425}{-5}\right)x=-\frac{3250}{-5}$$

-425 ni -5 ga bo'lish.

$$x^{2}+85x=-\frac{3250}{-5}$$

-3250 ni -5 ga bo'lish.

$$x^{2}+85x=650$$

85 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{85}{2} olish uchun. Keyin, \frac{85}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+85x+\left(\frac{85}{2}\right)^{2}=650+\left(\frac{85}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{85}{2} kvadratini chiqarish.

$$x^{2}+85x+\frac{7225}{4}=650+\frac{7225}{4}$$

650 ni \frac{7225}{4} ga qo'shish.

$$x^{2}+85x+\frac{7225}{4}=\frac{9825}{4}$$

x^{2}+85x+\frac{7225}{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.

$$\left(x+\frac{85}{2}\right)^{2}=\frac{9825}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{85}{2}\right)^{2}}=\sqrt{\frac{9825}{4}}$$

Qisqartirish.

$$x+\frac{85}{2}=\frac{5\sqrt{393}}{2} x+\frac{85}{2}=-\frac{5\sqrt{393}}{2}$$

Tenglamaning ikkala tarafidan \frac{85}{2} ni ayirish.

$$x=\frac{5\sqrt{393}-85}{2} x=\frac{-5\sqrt{393}-85}{2}$$