-15t^2-9t+45=0 eching

$$- 15 t ^ { 2 } - 9 t + 45 = 0$$

t uchun yechish

$$t = \frac{\sqrt{309} - 3}{10} \approx 1,457839583$$

$$t=\frac{-\sqrt{309}-3}{10}\approx -2,057839583$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$-15t^{2}-9t+45=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -15 ni a, -9 ni b va 45 ni c bilan almashtiring.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-15\right)\times 45}}{2\left(-15\right)}$$

-9 kvadratini chiqarish.

$$t=\frac{-\left(-9\right)±\sqrt{81-4\left(-15\right)\times 45}}{2\left(-15\right)}$$

-4 ni -15 marotabaga ko'paytirish.

$$t=\frac{-\left(-9\right)±\sqrt{81+60\times 45}}{2\left(-15\right)}$$

60 ni 45 marotabaga ko'paytirish.

$$t=\frac{-\left(-9\right)±\sqrt{81+2700}}{2\left(-15\right)}$$

81 ni 2700 ga qo'shish.

$$t=\frac{-\left(-9\right)±\sqrt{2781}}{2\left(-15\right)}$$

2781 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-9\right)±3\sqrt{309}}{2\left(-15\right)}$$

-9 ning teskarisi 9 ga teng.

$$t=\frac{9±3\sqrt{309}}{2\left(-15\right)}$$

2 ni -15 marotabaga ko'paytirish.

$$t=\frac{9±3\sqrt{309}}{-30}$$

t=\frac{9±3\sqrt{309}}{-30} tenglamasini yeching, bunda ± musbat. 9 ni 3\sqrt{309} ga qo'shish.

$$t=\frac{3\sqrt{309}+9}{-30}$$

9+3\sqrt{309} ni -30 ga bo'lish.

$$t=\frac{-\sqrt{309}-3}{10}$$

t=\frac{9±3\sqrt{309}}{-30} tenglamasini yeching, bunda ± manfiy. 9 dan 3\sqrt{309} ni ayirish.

$$t=\frac{9-3\sqrt{309}}{-30}$$

9-3\sqrt{309} ni -30 ga bo'lish.

$$t=\frac{\sqrt{309}-3}{10}$$

Tenglama yechildi.

$$t=\frac{-\sqrt{309}-3}{10} t=\frac{\sqrt{309}-3}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$-15t^{2}-9t+45=0$$

Tenglamaning ikkala tarafidan 45 ni ayirish.

$$-15t^{2}-9t+45-45=-45$$

O‘zidan 45 ayirilsa 0 qoladi.

$$-15t^{2}-9t=-45$$

Ikki tarafini -15 ga bo‘ling.

$$\frac{-15t^{2}-9t}{-15}=-\frac{45}{-15}$$

-15 ga bo'lish -15 ga ko'paytirishni bekor qiladi.

$$t^{2}+\left(-\frac{9}{-15}\right)t=-\frac{45}{-15}$$

\frac{-9}{-15} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}+\frac{3}{5}t=-\frac{45}{-15}$$

-45 ni -15 ga bo'lish.

$$t^{2}+\frac{3}{5}t=3$$

\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{10} olish uchun. Keyin, \frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}+\frac{3}{5}t+\left(\frac{3}{10}\right)^{2}=3+\left(\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{10} kvadratini chiqarish.

$$t^{2}+\frac{3}{5}t+\frac{9}{100}=3+\frac{9}{100}$$

3 ni \frac{9}{100} ga qo'shish.

$$t^{2}+\frac{3}{5}t+\frac{9}{100}=\frac{309}{100}$$

t^{2}+\frac{3}{5}t+\frac{9}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+\frac{3}{10}\right)^{2}=\frac{309}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+\frac{3}{10}\right)^{2}}=\sqrt{\frac{309}{100}}$$

Qisqartirish.

$$t+\frac{3}{10}=\frac{\sqrt{309}}{10} t+\frac{3}{10}=-\frac{\sqrt{309}}{10}$$

Tenglamaning ikkala tarafidan \frac{3}{10} ni ayirish.

$$t=\frac{\sqrt{309}-3}{10} t=\frac{-\sqrt{309}-3}{10}$$