3x^2-15x-6=3 eching

$$3 x ^ { 2 } - 15 x - 6 = 3$$

x uchun yechish

$$x = \frac{\sqrt{37} + 5}{2} \approx 5,541381265$$

$$x=\frac{5-\sqrt{37}}{2}\approx -0,541381265$$

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.

$$3x^{2}-15x-6=3$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$3x^{2}-15x-6-3=3-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$3x^{2}-15x-6-3=0$$

-6 dan 3 ni ayirish.

$$3x^{2}-15x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -15 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 3\left(-9\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-12\left(-9\right)}}{2\times 3}$$

-12 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225+108}}{2\times 3}$$

225 ni 108 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{333}}{2\times 3}$$

333 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-15\right)±3\sqrt{37}}{2\times 3}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±3\sqrt{37}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{15±3\sqrt{37}}{6}$$

x=\frac{15±3\sqrt{37}}{6} tenglamasini yeching, bunda ± musbat. 15 ni 3\sqrt{37} ga qo'shish.

$$x=\frac{3\sqrt{37}+15}{6}$$

15+3\sqrt{37} ni 6 ga bo'lish.

$$x=\frac{\sqrt{37}+5}{2}$$

x=\frac{15±3\sqrt{37}}{6} tenglamasini yeching, bunda ± manfiy. 15 dan 3\sqrt{37} ni ayirish.

$$x=\frac{15-3\sqrt{37}}{6}$$

15-3\sqrt{37} ni 6 ga bo'lish.

$$x=\frac{5-\sqrt{37}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{37}+5}{2} x=\frac{5-\sqrt{37}}{2}$$

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.

$$3x^{2}-15x-6=3$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-15x-6-\left(-6\right)=3-\left(-6\right)$$

O‘zidan -6 ayirilsa 0 qoladi.

$$3x^{2}-15x=3-\left(-6\right)$$

3 dan -6 ni ayirish.

$$3x^{2}-15x=9$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-15x}{3}=\frac{9}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{15}{3}\right)x=\frac{9}{3}$$

-15 ni 3 ga bo'lish.

$$x^{2}-5x=\frac{9}{3}$$

9 ni 3 ga bo'lish.

$$x^{2}-5x=3$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=3+\frac{25}{4}$$

3 ni \frac{25}{4} ga qo'shish.

$$x^{2}-5x+\frac{25}{4}=\frac{37}{4}$$

x^{2}-5x+\frac{25}{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.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{37}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{\sqrt{37}}{2} x-\frac{5}{2}=-\frac{\sqrt{37}}{2}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{37}+5}{2} x=\frac{5-\sqrt{37}}{2}$$