5=10t+5t^2 eching

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

t uchun yechish

$$t=\sqrt{2}-1\approx 0,414213562$$

$$t=-\sqrt{2}-1\approx -2,414213562$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

Ikkala tarafdan 5 ni ayirish.

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

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.

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

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 -5 ni c bilan almashtiring.

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

10 kvadratini chiqarish.

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

-4 ni 5 marotabaga ko'paytirish.

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

-20 ni -5 marotabaga ko'paytirish.

$$t=\frac{-10±\sqrt{100+100}}{2\times 5}$$

100 ni 100 ga qo'shish.

$$t=\frac{-10±\sqrt{200}}{2\times 5}$$

200 ning kvadrat ildizini chiqarish.

$$t=\frac{-10±10\sqrt{2}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$t=\frac{-10±10\sqrt{2}}{10}$$

t=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± musbat. -10 ni 10\sqrt{2} ga qo'shish.

$$t=\frac{10\sqrt{2}-10}{10}$$

-10+10\sqrt{2} ni 10 ga bo'lish.

$$t=\sqrt{2}-1$$

t=\frac{-10±10\sqrt{2}}{10} tenglamasini yeching, bunda ± manfiy. -10 dan 10\sqrt{2} ni ayirish.

$$t=\frac{-10\sqrt{2}-10}{10}$$

-10-10\sqrt{2} ni 10 ga bo'lish.

$$t=-\sqrt{2}-1$$

Tenglama yechildi.

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

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

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

Ikki tarafini 5 ga bo‘ling.

$$\frac{5t^{2}+10t}{5}=\frac{5}{5}$$

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

$$t^{2}+\frac{10}{5}t=\frac{5}{5}$$

10 ni 5 ga bo'lish.

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

5 ni 5 ga bo'lish.

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

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.

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

1 kvadratini chiqarish.

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

1 ni 1 ga qo'shish.

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

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$t+1=\sqrt{2} t+1=-\sqrt{2}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$