10-5t^2=10t-5 eching

$$10 - 5 t ^ { 2 } = 10 t - 5$$

t uchun yechish

$$t=-3$$

$$t=1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 10t ni ayirish.

$$10-5t^{2}-10t=-5$$

5 ni ikki tarafga qo’shing.

$$10-5t^{2}-10t+5=0$$

15 olish uchun 10 va 5'ni qo'shing.

$$15-5t^{2}-10t=0$$

Ikki tarafini 5 ga bo‘ling.

$$3-t^{2}-2t=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-t^{2}-2t+3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -t^{2}+at+bt+3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-2 ab=-3=-3$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=1 b=-3$$

-t^{2}-2t+3 ni \left(-t^{2}+t\right)+\left(-3t+3\right) sifatida qaytadan yozish.

$$\left(-t^{2}+t\right)+\left(-3t+3\right)$$

Birinchi guruhda t ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$t\left(-t+1\right)+3\left(-t+1\right)$$

Distributiv funktsiyasidan foydalangan holda -t+1 umumiy terminini chiqaring.

$$\left(-t+1\right)\left(t+3\right)$$

Tenglamani yechish uchun -t+1=0 va t+3=0 ni yeching.

$$t=1 t=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 10t ni ayirish.

$$10-5t^{2}-10t=-5$$

5 ni ikki tarafga qo’shing.

$$10-5t^{2}-10t+5=0$$

15 olish uchun 10 va 5'ni qo'shing.

$$15-5t^{2}-10t=0$$

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.

$$-5t^{2}-10t+15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, -10 ni b va 15 ni c bilan almashtiring.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-5\right)\times 15}}{2\left(-5\right)}$$

-10 kvadratini chiqarish.

$$t=\frac{-\left(-10\right)±\sqrt{100-4\left(-5\right)\times 15}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$t=\frac{-\left(-10\right)±\sqrt{100+20\times 15}}{2\left(-5\right)}$$

20 ni 15 marotabaga ko'paytirish.

$$t=\frac{-\left(-10\right)±\sqrt{100+300}}{2\left(-5\right)}$$

100 ni 300 ga qo'shish.

$$t=\frac{-\left(-10\right)±\sqrt{400}}{2\left(-5\right)}$$

400 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-10\right)±20}{2\left(-5\right)}$$

-10 ning teskarisi 10 ga teng.

$$t=\frac{10±20}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$t=\frac{10±20}{-10}$$

t=\frac{10±20}{-10} tenglamasini yeching, bunda ± musbat. 10 ni 20 ga qo'shish.

$$t=\frac{30}{-10}$$

30 ni -10 ga bo'lish.

$$t=-3$$

t=\frac{10±20}{-10} tenglamasini yeching, bunda ± manfiy. 10 dan 20 ni ayirish.

$$t=-\frac{10}{-10}$$

-10 ni -10 ga bo'lish.

$$t=1$$

Tenglama yechildi.

$$t=-3 t=1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 10t ni ayirish.

$$10-5t^{2}-10t=-5$$

Ikkala tarafdan 10 ni ayirish.

$$-5t^{2}-10t=-5-10$$

-15 olish uchun -5 dan 10 ni ayirish.

$$-5t^{2}-10t=-15$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5t^{2}-10t}{-5}=-\frac{15}{-5}$$

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

$$t^{2}+\left(-\frac{10}{-5}\right)t=-\frac{15}{-5}$$

-10 ni -5 ga bo'lish.

$$t^{2}+2t=-\frac{15}{-5}$$

-15 ni -5 ga bo'lish.

$$t^{2}+2t=3$$

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}+2t+1^{2}=3+1^{2}$$

1 kvadratini chiqarish.

$$t^{2}+2t+1=3+1$$

3 ni 1 ga qo'shish.

$$t^{2}+2t+1=4$$

t^{2}+2t+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+1\right)^{2}=4$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{4}$$

Qisqartirish.

$$t+1=2 t+1=-2$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$t=1 t=-3$$