5m^2-14m-15=0 eching

$$5 m ^ { 2 } - 14 m - 15 = 0$$

m uchun yechish

$$m = \frac{2 \sqrt{31} + 7}{5} \approx 3,627105745$$

$$m=\frac{7-2\sqrt{31}}{5}\approx -0,827105745$$

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.

$$5m^{2}-14m-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -14 ni b va -15 ni c bilan almashtiring.

$$m=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 5\left(-15\right)}}{2\times 5}$$

-14 kvadratini chiqarish.

$$m=\frac{-\left(-14\right)±\sqrt{196-4\times 5\left(-15\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$m=\frac{-\left(-14\right)±\sqrt{196-20\left(-15\right)}}{2\times 5}$$

-20 ni -15 marotabaga ko'paytirish.

$$m=\frac{-\left(-14\right)±\sqrt{196+300}}{2\times 5}$$

196 ni 300 ga qo'shish.

$$m=\frac{-\left(-14\right)±\sqrt{496}}{2\times 5}$$

496 ning kvadrat ildizini chiqarish.

$$m=\frac{-\left(-14\right)±4\sqrt{31}}{2\times 5}$$

-14 ning teskarisi 14 ga teng.

$$m=\frac{14±4\sqrt{31}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$m=\frac{14±4\sqrt{31}}{10}$$

m=\frac{14±4\sqrt{31}}{10} tenglamasini yeching, bunda ± musbat. 14 ni 4\sqrt{31} ga qo'shish.

$$m=\frac{4\sqrt{31}+14}{10}$$

14+4\sqrt{31} ni 10 ga bo'lish.

$$m=\frac{2\sqrt{31}+7}{5}$$

m=\frac{14±4\sqrt{31}}{10} tenglamasini yeching, bunda ± manfiy. 14 dan 4\sqrt{31} ni ayirish.

$$m=\frac{14-4\sqrt{31}}{10}$$

14-4\sqrt{31} ni 10 ga bo'lish.

$$m=\frac{7-2\sqrt{31}}{5}$$

Tenglama yechildi.

$$m=\frac{2\sqrt{31}+7}{5} m=\frac{7-2\sqrt{31}}{5}$$

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.

$$5m^{2}-14m-15=0$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$5m^{2}-14m-15-\left(-15\right)=-\left(-15\right)$$

O‘zidan -15 ayirilsa 0 qoladi.

$$5m^{2}-14m=-\left(-15\right)$$

0 dan -15 ni ayirish.

$$5m^{2}-14m=15$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5m^{2}-14m}{5}=\frac{15}{5}$$

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

$$m^{2}-\frac{14}{5}m=\frac{15}{5}$$

15 ni 5 ga bo'lish.

$$m^{2}-\frac{14}{5}m=3$$

-\frac{14}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{5} olish uchun. Keyin, -\frac{7}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$m^{2}-\frac{14}{5}m+\left(-\frac{7}{5}\right)^{2}=3+\left(-\frac{7}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{5} kvadratini chiqarish.

$$m^{2}-\frac{14}{5}m+\frac{49}{25}=3+\frac{49}{25}$$

3 ni \frac{49}{25} ga qo'shish.

$$m^{2}-\frac{14}{5}m+\frac{49}{25}=\frac{124}{25}$$

m^{2}-\frac{14}{5}m+\frac{49}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(m-\frac{7}{5}\right)^{2}=\frac{124}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(m-\frac{7}{5}\right)^{2}}=\sqrt{\frac{124}{25}}$$

Qisqartirish.

$$m-\frac{7}{5}=\frac{2\sqrt{31}}{5} m-\frac{7}{5}=-\frac{2\sqrt{31}}{5}$$

\frac{7}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$m=\frac{2\sqrt{31}+7}{5} m=\frac{7-2\sqrt{31}}{5}$$