17x^2-6x-15=0 eching

$$17 x ^ { 2 } - 6 x - 15 = 0$$

x uchun yechish

$$x = \frac{2 \sqrt{66} + 3}{17} \approx 1,132239812$$

$$x=\frac{3-2\sqrt{66}}{17}\approx -0,779298636$$

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.

$$17x^{2}-6x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 17 ni a, -6 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 17\left(-15\right)}}{2\times 17}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 17\left(-15\right)}}{2\times 17}$$

-4 ni 17 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-68\left(-15\right)}}{2\times 17}$$

-68 ni -15 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+1020}}{2\times 17}$$

36 ni 1020 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{1056}}{2\times 17}$$

1056 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±4\sqrt{66}}{2\times 17}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±4\sqrt{66}}{2\times 17}$$

2 ni 17 marotabaga ko'paytirish.

$$x=\frac{6±4\sqrt{66}}{34}$$

x=\frac{6±4\sqrt{66}}{34} tenglamasini yeching, bunda ± musbat. 6 ni 4\sqrt{66} ga qo'shish.

$$x=\frac{4\sqrt{66}+6}{34}$$

6+4\sqrt{66} ni 34 ga bo'lish.

$$x=\frac{2\sqrt{66}+3}{17}$$

x=\frac{6±4\sqrt{66}}{34} tenglamasini yeching, bunda ± manfiy. 6 dan 4\sqrt{66} ni ayirish.

$$x=\frac{6-4\sqrt{66}}{34}$$

6-4\sqrt{66} ni 34 ga bo'lish.

$$x=\frac{3-2\sqrt{66}}{17}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{66}+3}{17} x=\frac{3-2\sqrt{66}}{17}$$

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.

$$17x^{2}-6x-15=0$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$17x^{2}-6x-15-\left(-15\right)=-\left(-15\right)$$

O‘zidan -15 ayirilsa 0 qoladi.

$$17x^{2}-6x=-\left(-15\right)$$

0 dan -15 ni ayirish.

$$17x^{2}-6x=15$$

Ikki tarafini 17 ga bo‘ling.

$$\frac{17x^{2}-6x}{17}=\frac{15}{17}$$

17 ga bo'lish 17 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{6}{17}x=\frac{15}{17}$$

-\frac{6}{17} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{17} olish uchun. Keyin, -\frac{3}{17} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{6}{17}x+\left(-\frac{3}{17}\right)^{2}=\frac{15}{17}+\left(-\frac{3}{17}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{17} kvadratini chiqarish.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{15}{17}+\frac{9}{289}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{17} ni \frac{9}{289} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{6}{17}x+\frac{9}{289}=\frac{264}{289}$$

x^{2}-\frac{6}{17}x+\frac{9}{289} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{17}\right)^{2}=\frac{264}{289}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{17}\right)^{2}}=\sqrt{\frac{264}{289}}$$

Qisqartirish.

$$x-\frac{3}{17}=\frac{2\sqrt{66}}{17} x-\frac{3}{17}=-\frac{2\sqrt{66}}{17}$$

\frac{3}{17} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{66}+3}{17} x=\frac{3-2\sqrt{66}}{17}$$