65=-10t-5t^2 eching

$$65 = -10t-5 { t }^{ 2 }$$

t uchun yechish

$$t=-1+2\sqrt{3}i\approx -1+3,464101615i$$

$$t=-2\sqrt{3}i-1\approx -1-3,464101615i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-10t-5t^{2}=65$$

Ikkala tarafdan 65 ni ayirish.

$$-10t-5t^{2}-65=0$$

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.

$$-5t^{2}-10t-65=0$$

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 -65 ni c bilan almashtiring.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\left(-5\right)\left(-65\right)}}{2\left(-5\right)}$$

-10 kvadratini chiqarish.

$$t=\frac{-\left(-10\right)±\sqrt{100-4\left(-5\right)\left(-65\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$t=\frac{-\left(-10\right)±\sqrt{100+20\left(-65\right)}}{2\left(-5\right)}$$

20 ni -65 marotabaga ko'paytirish.

$$t=\frac{-\left(-10\right)±\sqrt{100-1300}}{2\left(-5\right)}$$

100 ni -1300 ga qo'shish.

$$t=\frac{-\left(-10\right)±\sqrt{-1200}}{2\left(-5\right)}$$

-1200 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-10\right)±20\sqrt{3}i}{2\left(-5\right)}$$

-10 ning teskarisi 10 ga teng.

$$t=\frac{10±20\sqrt{3}i}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$t=\frac{10±20\sqrt{3}i}{-10}$$

t=\frac{10±20\sqrt{3}i}{-10} tenglamasini yeching, bunda ± musbat. 10 ni 20i\sqrt{3} ga qo'shish.

$$t=\frac{10+20\sqrt{3}i}{-10}$$

10+20i\sqrt{3} ni -10 ga bo'lish.

$$t=-2\sqrt{3}i-1$$

t=\frac{10±20\sqrt{3}i}{-10} tenglamasini yeching, bunda ± manfiy. 10 dan 20i\sqrt{3} ni ayirish.

$$t=\frac{-20\sqrt{3}i+10}{-10}$$

10-20i\sqrt{3} ni -10 ga bo'lish.

$$t=-1+2\sqrt{3}i$$

Tenglama yechildi.

$$t=-2\sqrt{3}i-1 t=-1+2\sqrt{3}i$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-10t-5t^{2}=65$$

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

$$-5t^{2}-10t=65$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5t^{2}-10t}{-5}=\frac{65}{-5}$$

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

$$t^{2}+\left(-\frac{10}{-5}\right)t=\frac{65}{-5}$$

-10 ni -5 ga bo'lish.

$$t^{2}+2t=\frac{65}{-5}$$

65 ni -5 ga bo'lish.

$$t^{2}+2t=-13$$

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.

$$t^{2}+2t+1^{2}=-13+1^{2}$$

1 kvadratini chiqarish.

$$t^{2}+2t+1=-13+1$$

-13 ni 1 ga qo'shish.

$$t^{2}+2t+1=-12$$

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.

$$\left(t+1\right)^{2}=-12$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{-12}$$

Qisqartirish.

$$t+1=2\sqrt{3}i t+1=-2\sqrt{3}i$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$t=-1+2\sqrt{3}i t=-2\sqrt{3}i-1$$