x^2-15x-9=0 eching

$$x ^ { 2 } - 15 x - 9 = 0$$

x uchun yechish

$$x = \frac{3 \sqrt{29} + 15}{2} \approx 15,577747211$$

$$x=\frac{15-3\sqrt{29}}{2}\approx -0,577747211$$

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.

$$x^{2}-15x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -15 ni b va -9 ni c bilan almashtiring.

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

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-9\right)}}{2}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225+36}}{2}$$

225 ni 36 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{261}}{2}$$

261 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-15\right)±3\sqrt{29}}{2}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±3\sqrt{29}}{2}$$

x=\frac{15±3\sqrt{29}}{2} tenglamasini yeching, bunda ± musbat. 15 ni 3\sqrt{29} ga qo'shish.

$$x=\frac{3\sqrt{29}+15}{2}$$

x=\frac{15±3\sqrt{29}}{2} tenglamasini yeching, bunda ± manfiy. 15 dan 3\sqrt{29} ni ayirish.

$$x=\frac{15-3\sqrt{29}}{2}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{29}+15}{2} x=\frac{15-3\sqrt{29}}{2}$$

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.

$$x^{2}-15x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}-15x-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$x^{2}-15x=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$x^{2}-15x=9$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=9+\frac{225}{4}$$

9 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=\frac{261}{4}$$

x^{2}-15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{261}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{261}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{3\sqrt{29}}{2} x-\frac{15}{2}=-\frac{3\sqrt{29}}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{29}+15}{2} x=\frac{15-3\sqrt{29}}{2}$$